Ideals in multivariate polynomial rings

Sage has a powerful system to compute with multivariate polynomial rings. Most algorithms dealing with these ideals are centered on the computation of Groebner bases. Sage mainly uses Singular to implement this functionality. Singular is widely regarded as the best open-source system for Groebner basis calculation in multivariate polynomial rings over fields.

EXAMPLES:

We compute a Groebner basis for some given ideal. The type returned by the groebner_basis method is PolynomialSequence, i.e. it is not a MPolynomialIdeal:

sage: x,y,z = QQ['x,y,z'].gens()
sage: I = ideal(x^5 + y^4 + z^3 - 1,  x^3 + y^3 + z^2 - 1)
sage: B = I.groebner_basis()
sage: type(B)
<class 'sage.rings.polynomial.multi_polynomial_sequence.PolynomialSequence_generic'>

Groebner bases can be used to solve the ideal membership problem:

sage: f,g,h = B
sage: (2*x*f + g).reduce(B)
0

sage: (2*x*f + g) in I
True

sage: (2*x*f + 2*z*h + y^3).reduce(B)
y^3

sage: (2*x*f + 2*z*h + y^3) in I
False

We compute a Groebner basis for Cyclic 6, which is a standard benchmark and test ideal.

sage: R.<x,y,z,t,u,v> = QQ['x,y,z,t,u,v']
sage: I = sage.rings.ideal.Cyclic(R,6)
sage: B = I.groebner_basis()
sage: len(B)
45

We compute in a quotient of a polynomial ring over \(\ZZ/17\ZZ\):

sage: R.<x,y> = ZZ[]
sage: S.<a,b> = R.quotient((x^2 + y^2, 17))
sage: S
Quotient of Multivariate Polynomial Ring in x, y over Integer Ring
by the ideal (x^2 + y^2, 17)

sage: a^2 + b^2 == 0
True
sage: a^3 - b^2
-a*b^2 - b^2

Note that the result of a computation is not necessarily reduced:

sage: (a+b)^17
256*a*b^16 + 256*b^17
sage: S(17) == 0
True

Or we can work with \(\ZZ/17\ZZ\) directly:

sage: R.<x,y> = Zmod(17)[]
sage: S.<a,b> = R.quotient((x^2 + y^2,))
sage: S
Quotient of Multivariate Polynomial Ring in x, y over Ring of
integers modulo 17 by the ideal (x^2 + y^2)

sage: a^2 + b^2 == 0
True
sage: a^3 - b^2 == -a*b^2 - b^2 == 16*a*b^2 + 16*b^2
True
sage: (a+b)^17
a*b^16 + b^17
sage: S(17) == 0
True

Working with a polynomial ring over \(\ZZ\):

sage: R.<x,y,z,w> = ZZ[]
sage: I = ideal(x^2 + y^2 - z^2 - w^2, x-y)
sage: J = I^2
sage: J.groebner_basis()
[4*y^4 - 4*y^2*z^2 + z^4 - 4*y^2*w^2 + 2*z^2*w^2 + w^4,
 2*x*y^2 - 2*y^3 - x*z^2 + y*z^2 - x*w^2 + y*w^2,
 x^2 - 2*x*y + y^2]

sage: y^2 - 2*x*y + x^2 in J
True
sage: 0 in J
True

We do a Groebner basis computation over a number field:

sage: K.<zeta> = CyclotomicField(3)
sage: R.<x,y,z> = K[]; R
Multivariate Polynomial Ring in x, y, z over Cyclotomic Field of order 3 and degree 2

sage: i = ideal(x - zeta*y + 1, x^3 - zeta*y^3); i
Ideal (x + (-zeta)*y + 1, x^3 + (-zeta)*y^3) of Multivariate
Polynomial Ring in x, y, z over Cyclotomic Field of order 3 and degree 2

sage: i.groebner_basis()
[y^3 + (2*zeta + 1)*y^2 + (zeta - 1)*y + (-1/3*zeta - 2/3), x + (-zeta)*y + 1]

sage: S = R.quotient(i); S
Quotient of Multivariate Polynomial Ring in x, y, z over
Cyclotomic Field of order 3 and degree 2 by the ideal (x +
(-zeta)*y + 1, x^3 + (-zeta)*y^3)

sage: S.0  - zeta*S.1
-1
sage: S.0^3 - zeta*S.1^3
0

Two examples from the Mathematica documentation (done in Sage):

We compute a Groebner basis:

sage: R.<x,y> = PolynomialRing(QQ, order='lex')
sage: ideal(x^2 - 2*y^2, x*y - 3).groebner_basis()
[x - 2/3*y^3, y^4 - 9/2]

We show that three polynomials have no common root:

sage: R.<x,y> = QQ[]
sage: ideal(x+y, x^2 - 1, y^2 - 2*x).groebner_basis()
[1]

The next example shows how we can use Groebner bases over \(\ZZ\) to find the primes modulo which a system of equations has a solution, when the system has no solutions over the rationals.

We first form a certain ideal \(I\) in \(\ZZ[x, y, z]\), and note that the Groebner basis of \(I\) over \(\QQ\) contains 1, so there are no solutions over \(\QQ\) or an algebraic closure of it (this is not surprising as there are 4 equations in 3 unknowns).

sage: P.<x,y,z> = PolynomialRing(ZZ,order='lex')
sage: I = ideal(-y^2 - 3*y + z^2 + 3, -2*y*z + z^2 + 2*z + 1, \
                x*z + y*z + z^2, -3*x*y + 2*y*z + 6*z^2)
sage: I.change_ring(P.change_ring(QQ)).groebner_basis()
[1]

However, when we compute the Groebner basis of I (defined over \(\ZZ\)), we note that there is a certain integer in the ideal which is not 1.

sage: I.groebner_basis()
[x + y + 57119*z + 4, y^2 + 3*y + 17220, y*z + y + 26532, 2*y + 158864, z^2 + 17223, 2*z + 41856, 164878]

Now for each prime \(p\) dividing this integer 164878, the Groebner basis of I modulo \(p\) will be non-trivial and will thus give a solution of the original system modulo \(p\).

sage: factor(164878)
2 * 7 * 11777

sage: I.change_ring(P.change_ring( GF(2) )).groebner_basis()
[x + y + z, y^2 + y, y*z + y, z^2 + 1]
sage: I.change_ring(P.change_ring( GF(7) )).groebner_basis()
[x - 1, y + 3, z - 2]
sage: I.change_ring(P.change_ring( GF(11777 ))).groebner_basis()
[x + 5633, y - 3007, z - 2626]

The Groebner basis modulo any product of the prime factors is also non-trivial:

sage: I.change_ring(P.change_ring( IntegerModRing(2*7) )).groebner_basis()
[x + 9*y + 13*z, y^2 + 3*y, y*z + 7*y + 6, 2*y + 6, z^2 + 3, 2*z + 10]

Modulo any other prime the Groebner basis is trivial so there are no other solutions. For example:

sage: I.change_ring( P.change_ring( GF(3) ) ).groebner_basis()
[1]

Note

Sage distinguishes between lists or sequences of polynomials and ideals. Thus an ideal is not identified with a particular set of generators. For sequences of multivariate polynomials see sage.rings.polynomial.multi_polynomial_sequence.PolynomialSequence_generic.

AUTHORS:

• William Stein: initial version

• Kiran S. Kedlaya (2006-02-12): added Macaulay2 analogues of some Singular features

• Martin Albrecht (2007,2008): refactoring, many Singular related functions, added plot()

• Martin Albrecht (2009): added Groebner basis over rings functionality from Singular 3.1

• John Perry (2012): bug fixing equality & containment of ideals

class sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal(ring, gens, coerce=True)

Bases: sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal_singular_repr, sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal_macaulay2_repr, sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal_magma_repr, sage.rings.ideal.Ideal_generic

Create an ideal in a multivariate polynomial ring.

INPUT:

• ring - the ring the ideal is defined in

• gens - a list of generators for the ideal

• coerce - coerce elements to the ring ring?

EXAMPLES:

sage: R.<x,y> = PolynomialRing(IntegerRing(), 2, order='lex')
sage: R.ideal([x, y])
Ideal (x, y) of Multivariate Polynomial Ring in x, y over Integer Ring
sage: R.<x0,x1> = GF(3)[]
sage: R.ideal([x0^2, x1^3])
Ideal (x0^2, x1^3) of Multivariate Polynomial Ring in x0, x1 over Finite Field of size 3

basis

Shortcut to gens().

EXAMPLES:

sage: P.<x,y> = PolynomialRing(QQ,2)
sage: I = Ideal([x,y+1])
sage: I.basis
[x, y + 1]

change_ring(P)

Return the ideal I in P spanned by the generators \(g_1, ..., g_n\) of self as returned by self.gens().

INPUT:

• P - a multivariate polynomial ring

EXAMPLES:

sage: P.<x,y,z> = PolynomialRing(QQ,3,order='lex')
sage: I = sage.rings.ideal.Cyclic(P)
sage: I
Ideal (x + y + z, x*y + x*z + y*z, x*y*z - 1) of
Multivariate Polynomial Ring in x, y, z over Rational Field

sage: I.groebner_basis()
[x + y + z, y^2 + y*z + z^2, z^3 - 1]

sage: Q.<x,y,z> = P.change_ring(order='degrevlex'); Q
Multivariate Polynomial Ring in x, y, z over Rational Field
sage: Q.term_order()
Degree reverse lexicographic term order

sage: J = I.change_ring(Q); J
Ideal (x + y + z, x*y + x*z + y*z, x*y*z - 1) of
Multivariate Polynomial Ring in x, y, z over Rational Field

sage: J.groebner_basis()
[z^3 - 1, y^2 + y*z + z^2, x + y + z]

degree_of_semi_regularity()

Return the degree of semi-regularity of this ideal under the assumption that it is semi-regular.

Let \(\{f_1, ... , f_m\} \subset K[x_1 , ... , x_n]\) be homogeneous polynomials of degrees \(d_1,... ,d_m\) respectively. This sequence is semi-regular if:

• \(\{f_1, ... , f_m\} \neq K[x_1 , ... , x_n]\)

• for all \(1 \leq i \leq m\) and \(g \in K[x_1,\dots,x_n]\): \(deg(g \cdot pi ) < D\) and \(g \cdot f_i \in <f_1 , \dots , f_{i-1}>\) implies that \(g \in <f_1, ..., f_{i-1}>\) where \(D\) is the degree of regularity.

This notion can be extended to affine polynomials by considering their homogeneous components of highest degree.

The degree of regularity of a semi-regular sequence \(f_1, ...,f_m\) of respective degrees \(d_1,...,d_m\) is given by the index of the first non-positive coefficient of:

\(\sum c_k z^k = \frac{\prod (1 - z^{d_i})}{(1-z)^n}\)

EXAMPLES:

We consider a homogeneous example:

sage: n = 8
sage: K = GF(127)
sage: P = PolynomialRing(K,n,'x')
sage: s = [K.random_element() for _ in range(n)]
sage: L = []
sage: for i in range(2*n):
....:     f = P.random_element(degree=2, terms=binomial(n,2))
....:     f -= f(*s)
....:     L.append(f.homogenize())
sage: I = Ideal(L)
sage: I.degree_of_semi_regularity()
4

From this, we expect a Groebner basis computation to reach at most degree 4. For homogeneous systems this is equivalent to the largest degree in the Groebner basis:

sage: max(f.degree() for f in I.groebner_basis())
4

We increase the number of polynomials and observe a decrease the degree of regularity:

sage: for i in range(2*n):
....:     f = P.random_element(degree=2, terms=binomial(n,2))
....:     f -= f(*s)
....:     L.append(f.homogenize())
sage: I = Ideal(L)
sage: I.degree_of_semi_regularity()
3

sage: max(f.degree() for f in I.groebner_basis())
3

The degree of regularity approaches 2 for quadratic systems as the number of polynomials approaches \(n^2\):

sage: for i in range((n-4)*n):
....:     f = P.random_element(degree=2, terms=binomial(n,2))
....:     f -= f(*s)
....:     L.append(f.homogenize())
sage: I = Ideal(L)
sage: I.degree_of_semi_regularity()
2

sage: max(f.degree() for f in I.groebner_basis())
2

Note

It is unknown whether semi-regular sequences exist. However, it is expected that random systems are semi-regular sequences. For more details about semi-regular sequences see [BFS2004].

gens()

Return a set of generators / a basis of this ideal. This is usually the set of generators provided during object creation.

EXAMPLES:

sage: P.<x,y> = PolynomialRing(QQ,2)
sage: I = Ideal([x,y+1]); I
Ideal (x, y + 1) of Multivariate Polynomial Ring in x, y over Rational Field
sage: I.gens()
[x, y + 1]

groebner_basis(algorithm='', deg_bound=None, mult_bound=None, prot=False, *args, **kwds)

Return the reduced Groebner basis of this ideal.

A Groebner basis \(g_1,...,g_n\) for an ideal \(I\) is a generating set such that \(<LM(g_i)> = LM(I)\), i.e., the leading monomial ideal of \(I\) is spanned by the leading terms of \(g_1,...,g_n\). Groebner bases are the key concept in computational ideal theory in multivariate polynomial rings which allows a variety of problems to be solved.

Additionally, a reduced Groebner basis \(G\) is a unique representation for the ideal \(<G>\) with respect to the chosen monomial ordering.

INPUT:

• algorithm - determines the algorithm to use, see below

  for available algorithms.

• deg_bound - only compute to degree deg_bound, that is, ignore all S-polynomials of higher degree. (default: None)

• mult_bound - the computation is stopped if the ideal is zero-dimensional in a ring with local ordering and its multiplicity is lower than mult_bound. Singular only. (default: None)

• prot - if set to True the computation protocol of the underlying implementation is printed. If an algorithm from the singular: or magma: family is used, prot may also be sage in which case the output is parsed and printed in a common format where the amount of information printed can be controlled via calls to set_verbose().

• *args - additional parameters passed to the respective

  implementations

• **kwds - additional keyword parameters passed to the

  respective implementations

ALGORITHMS:

‘’

autoselect (default)

‘singular:groebner’

Singular’s groebner command

‘singular:std’

Singular’s std command

‘singular:stdhilb’

Singular’s stdhib command

‘singular:stdfglm’

Singular’s stdfglm command

‘singular:slimgb’

Singular’s slimgb command

‘libsingular:groebner’

libSingular’s groebner command

‘libsingular:std’

libSingular’s std command

‘libsingular:slimgb’

libSingular’s slimgb command

‘libsingular:stdhilb’

libSingular’s stdhib command

‘libsingular:stdfglm’

libSingular’s stdfglm command

‘toy:buchberger’

Sage’s toy/educational buchberger without Buchberger criteria

‘toy:buchberger2’

Sage’s toy/educational buchberger with Buchberger criteria

‘toy:d_basis’

Sage’s toy/educational algorithm for computation over PIDs

‘macaulay2:gb’

Macaulay2’s gb command (if available)

‘macaulay2:f4’

Macaulay2’s GroebnerBasis command with the strategy “F4” (if available)

‘macaulay2:mgb’

Macaulay2’s GroebnerBasis command with the strategy “MGB” (if available)

‘magma:GroebnerBasis’

Magma’s Groebnerbasis command (if available)

‘ginv:TQ’, ‘ginv:TQBlockHigh’, ‘ginv:TQBlockLow’ and ‘ginv:TQDegree’

One of GINV’s implementations (if available)

‘giac:gbasis’

Giac’s gbasis command (if available)

If only a system is given - e.g. ‘magma’ - the default algorithm is chosen for that system.

Note

The Singular and libSingular versions of the respective algorithms are identical, but the former calls an external Singular process while the latter calls a C function, i.e. the calling overhead is smaller. However, the libSingular interface does not support pretty printing of computation protocols.

EXAMPLES:

Consider Katsura-3 over \(\QQ\) with lexicographical term ordering. We compute the reduced Groebner basis using every available implementation and check their equality.

sage: P.<a,b,c> = PolynomialRing(QQ,3, order='lex')
sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
sage: I.groebner_basis()
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
sage: I.groebner_basis('libsingular:groebner')
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
sage: I.groebner_basis('libsingular:std')
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
sage: I.groebner_basis('libsingular:stdhilb')
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
sage: I.groebner_basis('libsingular:stdfglm')
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
sage: I.groebner_basis('libsingular:slimgb')
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

Although Giac does support lexicographical ordering, we use degree reverse lexicographical ordering here, in order to test against trac ticket #21884:

sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
sage: J = I.change_ring(P.change_ring(order='degrevlex'))
sage: gb = J.groebner_basis('giac') # random
sage: gb
[c^3 - 79/210*c^2 + 1/30*b + 1/70*c, b^2 - 3/5*c^2 - 1/5*b + 1/5*c, b*c + 6/5*c^2 - 1/10*b - 2/5*c, a + 2*b + 2*c - 1]

sage: J.groebner_basis.set_cache(gb)
sage: ideal(J.transformed_basis()).change_ring(P).interreduced_basis()  # testing trac 21884
...[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

Giac’s gbasis over \(\QQ\) can benefit from a probabilistic lifting and multi threaded operations:

sage: A9=PolynomialRing(QQ,9,'x')
sage: I9=sage.rings.ideal.Katsura(A9)
sage: print("possible output from giac", flush=True); I9.groebner_basis("giac",proba_epsilon=1e-7) # long time (3s)
possible output...
Polynomial Sequence with 143 Polynomials in 9 Variables

The list of available Giac options is provided at sage.libs.giac.groebner_basis().

Note that toy:buchberger does not return the reduced Groebner basis,

sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
sage: gb = I.groebner_basis('toy:buchberger')
sage: gb.is_groebner()
True
sage: gb == gb.reduced()
False

but that toy:buchberger2 does.

sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
sage: gb = I.groebner_basis('toy:buchberger2'); gb
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]
sage: gb == gb.reduced()
True

Here we use Macaulay2 with three different strategies over a finite field.

sage: R.<a,b,c> = PolynomialRing(GF(101), 3)
sage: I = sage.rings.ideal.Katsura(R,3) # regenerate to prevent caching
sage: I.groebner_basis('macaulay2:gb')  # optional - macaulay2
[c^3 + 28*c^2 - 37*b + 13*c, b^2 - 41*c^2 + 20*b - 20*c, b*c - 19*c^2 + 10*b + 40*c, a + 2*b + 2*c - 1]

sage: I = sage.rings.ideal.Katsura(R,3) # regenerate to prevent caching
sage: I.groebner_basis('macaulay2:f4')  # optional - macaulay2
[c^3 + 28*c^2 - 37*b + 13*c, b^2 - 41*c^2 + 20*b - 20*c, b*c - 19*c^2 + 10*b + 40*c, a + 2*b + 2*c - 1]

sage: I = sage.rings.ideal.Katsura(R,3) # regenerate to prevent caching
sage: I.groebner_basis('macaulay2:mgb') # optional - macaulay2
[c^3 + 28*c^2 - 37*b + 13*c, b^2 - 41*c^2 + 20*b - 20*c, b*c - 19*c^2 + 10*b + 40*c, a + 2*b + 2*c - 1]

sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
sage: I.groebner_basis('magma:GroebnerBasis') # optional - magma
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

Singular and libSingular can compute Groebner basis with degree restrictions.

sage: R.<x,y> = QQ[]
sage: I = R*[x^3+y^2,x^2*y+1]
sage: I.groebner_basis(algorithm='singular')
[x^3 + y^2, x^2*y + 1, y^3 - x]
sage: I.groebner_basis(algorithm='singular',deg_bound=2)
[x^3 + y^2, x^2*y + 1]
sage: I.groebner_basis()
[x^3 + y^2, x^2*y + 1, y^3 - x]
sage: I.groebner_basis(deg_bound=2)
[x^3 + y^2, x^2*y + 1]

A protocol is printed, if the verbosity level is at least 2, or if the argument prot is provided. Historically, the protocol did not appear during doctests, so, we skip the examples with protocol output.

sage: from sage.misc.verbose import set_verbose
sage: set_verbose(2)
sage: I = R*[x^3+y^2,x^2*y+1]
sage: I.groebner_basis()  # not tested
std in (QQ),(x,y),(dp(2),C)
[...:2]3ss4s6
(S:2)--
product criterion:1 chain criterion:0
[x^3 + y^2, x^2*y + 1, y^3 - x]
sage: I.groebner_basis(prot=False)
std in (QQ),(x,y),(dp(2),C)
[...:2]3ss4s6
(S:2)--
product criterion:1 chain criterion:0
[x^3 + y^2, x^2*y + 1, y^3 - x]
sage: set_verbose(0)
sage: I.groebner_basis(prot=True)  # not tested
std in (QQ),(x,y),(dp(2),C)
[...:2]3ss4s6
(S:2)--
product criterion:1 chain criterion:0
[x^3 + y^2, x^2*y + 1, y^3 - x]

The list of available options is provided at LibSingularOptions.

Note that Groebner bases over \(\ZZ\) can also be computed.

sage: P.<a,b,c> = PolynomialRing(ZZ,3)
sage: I = P * (a + 2*b + 2*c - 1, a^2 - a + 2*b^2 + 2*c^2, 2*a*b + 2*b*c - b)
sage: I.groebner_basis()
[b^3 + b*c^2 + 12*c^3 + b^2 + b*c - 4*c^2,
 2*b*c^2 - 6*c^3 - b^2 - b*c + 2*c^2,
 42*c^3 + b^2 + 2*b*c - 14*c^2 + b,
 2*b^2 + 6*b*c + 6*c^2 - b - 2*c,
 10*b*c + 12*c^2 - b - 4*c,
 a + 2*b + 2*c - 1]

sage: I.groebner_basis('macaulay2') # optional - macaulay2
[b^3 + b*c^2 + 12*c^3 + b^2 + b*c - 4*c^2,
 2*b*c^2 - 6*c^3 + b^2 + 5*b*c + 8*c^2 - b - 2*c,
 42*c^3 + b^2 + 2*b*c - 14*c^2 + b,
 2*b^2 - 4*b*c - 6*c^2 + 2*c, 10*b*c + 12*c^2 - b - 4*c,
 a + 2*b + 2*c - 1]

Groebner bases over \(\ZZ/n\ZZ\) are also supported:

sage: P.<a,b,c> = PolynomialRing(Zmod(1000),3)
sage: I = P * (a + 2*b + 2*c - 1, a^2 - a + 2*b^2 + 2*c^2, 2*a*b + 2*b*c - b)
sage: I.groebner_basis()
[b*c^2 + 732*b*c + 808*b,
 2*c^3 + 884*b*c + 666*c^2 + 320*b,
 b^2 + 438*b*c + 281*b,
 5*b*c + 156*c^2 + 112*b + 948*c,
 50*c^2 + 600*b + 650*c,
 a + 2*b + 2*c + 999,
 125*b]

sage: R.<x,y,z> = PolynomialRing(Zmod(2233497349584))
sage: I = R.ideal([z*(x-3*y), 3^2*x^2-y*z, z^2+y^2])
sage: I.groebner_basis()
[2*z^4, y*z^2 + 81*z^3, 248166372176*z^3, 9*x^2 - y*z, y^2 + z^2, x*z +
2233497349581*y*z, 248166372176*y*z]

Sage also supports local orderings:

sage: P.<x,y,z> = PolynomialRing(QQ,3,order='negdegrevlex')
sage: I = P * (  x*y*z + z^5, 2*x^2 + y^3 + z^7, 3*z^5 +y ^5 )
sage: I.groebner_basis()
[x^2 + 1/2*y^3, x*y*z + z^5, y^5 + 3*z^5, y^4*z - 2*x*z^5, z^6]

We can represent every element in the ideal as a combination of the generators using the lift() method:

sage: P.<x,y,z> = PolynomialRing(QQ,3)
sage: I = P * ( x*y*z + z^5, 2*x^2 + y^3 + z^7, 3*z^5 +y ^5 )
sage: J = Ideal(I.groebner_basis())
sage: f = sum(P.random_element(terms=2)*f for f in I.gens())
sage: f                       # random
1/2*y^2*z^7 - 1/4*y*z^8 + 2*x*z^5 + 95*z^6 + 1/2*y^5 - 1/4*y^4*z + x^2*y^2 + 3/2*x^2*y*z + 95*x*y*z^2
sage: f.lift(I.gens())        # random
[2*x + 95*z, 1/2*y^2 - 1/4*y*z, 0]
sage: l = f.lift(J.gens()); l # random
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1/2*y^2 + 1/4*y*z, 1/2*y^2*z^2 - 1/4*y*z^3 + 2*x + 95*z]
sage: sum(map(mul, zip(l,J.gens()))) == f
True

Groebner bases over fraction fields of polynomial rings are also supported:

sage: P.<t> = QQ[]
sage: F = Frac(P)
sage: R.<X,Y,Z> = F[]
sage: I = Ideal([f + P.random_element() for f in sage.rings.ideal.Katsura(R).gens()])
sage: I.groebner_basis()
[Z^3 + (-79/105*t - 79/70)*Z^2 + (2/15*t^2 - 74/315*t + 94/105)*Y + (2/35*t^2 + 194/315*t + 1/105)*Z - 4/105*t^2 - 17/210*t - 1/28, Y^2 + (-3/5)*Z^2 + (-2/5*t - 3/5)*Y + (2/5*t + 3/5)*Z - 4/15*t + 1/2, Y*Z + 6/5*Z^2 + (-1/5*t - 3/10)*Y + (-4/5*t - 6/5)*Z + 8/15*t - 1/2, X + 2*Y + 2*Z - t - 2]

In cases where a characteristic cannot be determined, we use a toy implementation of Buchberger’s algorithm (see trac ticket #6581):

sage: R.<a,b> = QQ[]; I = R.ideal(a^2+b^2-1)
sage: Q = QuotientRing(R,I); K = Frac(Q)
sage: R2.<x,y> = K[]; J = R2.ideal([(a^2+b^2)*x + y, x+y])
sage: J.groebner_basis()
verbose 0 (...: multi_polynomial_ideal.py, groebner_basis) Warning: falling back to very slow toy implementation.
[x + y]

ALGORITHM:

Uses Singular, Magma (if available), Macaulay2 (if available), Giac (if available), or a toy implementation.

groebner_fan(is_groebner_basis=False, symmetry=None, verbose=False)

Return the Groebner fan of this ideal.

The base ring must be \(\QQ\) or a finite field \(\GF{p}\) of with \(p \leq 32749\).

EXAMPLES:

sage: P.<x,y> = PolynomialRing(QQ)
sage: i = ideal(x^2 - y^2 + 1)
sage: g = i.groebner_fan()
sage: g.reduced_groebner_bases()
[[x^2 - y^2 + 1], [-x^2 + y^2 - 1]]

INPUT:

• is_groebner_basis - bool (default False). if True, then I.gens() must be a Groebner basis with respect to the standard degree lexicographic term order.

• symmetry - default: None; if not None, describes symmetries of the ideal

• verbose - default: False; if True, printout useful info during computations

homogenize(var='h')

Return homogeneous ideal spanned by the homogeneous polynomials generated by homogenizing the generators of this ideal.

INPUT:

• h - variable name or variable in cover ring (default: ‘h’)

EXAMPLES:

sage: P.<x,y,z> = PolynomialRing(GF(2))
sage: I = Ideal([x^2*y + z + 1, x + y^2 + 1]); I
Ideal (x^2*y + z + 1, y^2 + x + 1) of Multivariate
Polynomial Ring in x, y, z over Finite Field of size 2

sage: I.homogenize()
Ideal (x^2*y + z*h^2 + h^3, y^2 + x*h + h^2) of
Multivariate Polynomial Ring in x, y, z, h over Finite
Field of size 2

sage: I.homogenize(y)
Ideal (x^2*y + y^3 + y^2*z, x*y) of Multivariate
Polynomial Ring in x, y, z over Finite Field of size 2

       sage: I = Ideal([x^2*y + z^3 + y^2*x, x + y^2 + 1])
sage: I.homogenize()
Ideal (x^2*y + x*y^2 + z^3, y^2 + x*h + h^2) of
Multivariate Polynomial Ring in x, y, z, h over Finite
Field of size 2

is_homogeneous()

Return True if this ideal is spanned by homogeneous polynomials, i.e. if it is a homogeneous ideal.

EXAMPLES:

sage: P.<x,y,z> = PolynomialRing(QQ,3)
sage: I = sage.rings.ideal.Katsura(P)
sage: I
Ideal (x + 2*y + 2*z - 1, x^2 + 2*y^2 + 2*z^2 - x, 2*x*y +
2*y*z - y) of Multivariate Polynomial Ring in x, y, z over
Rational Field

sage: I.is_homogeneous()
False

sage: J = I.homogenize()
sage: J
Ideal (x + 2*y + 2*z - h, x^2 + 2*y^2 + 2*z^2 - x*h, 2*x*y
+ 2*y*z - y*h) of Multivariate Polynomial Ring in x, y, z,
h over Rational Field

sage: J.is_homogeneous()
True

plot(*args, **kwds)

Plot the real zero locus of this principal ideal.

INPUT:

• self - a principal ideal in 2 variables

• algorithm - set this to ‘surf’ if you want ‘surf’ to

  plot the ideal (default: None)

• *args - optional tuples (variable, minimum, maximum)

  for plotting dimensions

• **kwds - optional keyword arguments passed on to

  implicit_plot

EXAMPLES:

Implicit plotting in 2-d:

sage: R.<x,y> = PolynomialRing(QQ,2)
sage: I = R.ideal([y^3 - x^2])
sage: I.plot()                         # cusp
Graphics object consisting of 1 graphics primitive

sage: I = R.ideal([y^2 - x^2 - 1])
sage: I.plot((x,-3, 3), (y, -2, 2))  # hyperbola
Graphics object consisting of 1 graphics primitive

sage: I = R.ideal([y^2 + x^2*(1/4) - 1])
sage: I.plot()                         # ellipse
Graphics object consisting of 1 graphics primitive

sage: I = R.ideal([y^2-(x^2-1)*(x-2)])
sage: I.plot()                         # elliptic curve
Graphics object consisting of 1 graphics primitive

sage: f = ((x+3)^3 + 2*(x+3)^2 - y^2)*(x^3 - y^2)*((x-3)^3-2*(x-3)^2-y^2)
sage: I = R.ideal(f)
sage: I.plot()                         # the Singular logo
Graphics object consisting of 1 graphics primitive

sage: R.<x,y> = PolynomialRing(QQ,2)
sage: I = R.ideal([x - 1])
sage: I.plot((y, -2, 2))               # vertical line
Graphics object consisting of 1 graphics primitive

sage: I = R.ideal([-x^2*y + 1])
sage: I.plot()                         # blow up
Graphics object consisting of 1 graphics primitive

random_element(degree, compute_gb=False, *args, **kwds)

Return a random element in this ideal as \(r = \sum h_i·f_i\).

INPUT:

• compute_gb - if True then a Gröbner basis is computed first and \(f_i\) are the elements in the Gröbner basis. Otherwise whatever basis is returned by self.gens() is used.

• *args and **kwds are passed to R.random_element() with R = self.ring().

EXAMPLES:

We compute a uniformly random element up to the provided degree.

sage: P.<x,y,z> = GF(127)[]
sage: I = sage.rings.ideal.Katsura(P)
sage: I.random_element(degree=4, compute_gb=True, terms=infinity)
34*x^4 - 33*x^3*y + 45*x^2*y^2 - 51*x*y^3 - 55*y^4 + 43*x^3*z ... - 28*y - 33*z + 45

Note that sampling uniformly at random from the ideal at some large enough degree is equivalent to computing a Gröbner basis. We give an example showing how to compute a Gröbner basis if we can sample uniformly at random from an ideal:

sage: n = 3; d = 4
sage: P = PolynomialRing(GF(127), n, 'x')
sage: I = sage.rings.ideal.Cyclic(P)

1. We sample \(n^d\) uniformly random elements in the ideal:

   sage: F = Sequence(I.random_element(degree=d, compute_gb=True, terms=infinity) for _ in range(n^d))
   

2. We linearize and compute the echelon form:

   sage: A,v = F.coefficient_matrix()
   sage: A.echelonize()
   

3. The result is the desired Gröbner basis:

   sage: G = Sequence((A*v).list())
   sage: G.is_groebner()
   True
   sage: Ideal(G) == I
   True
   

We return some element in the ideal with no guarantee on the distribution:

sage: P = PolynomialRing(GF(127), 10, 'x')
sage: I = sage.rings.ideal.Katsura(P)
sage: f = I.random_element(degree=3)
sage: f  # random
-25*x0^2*x1 + 14*x1^3 + 57*x0*x1*x2 + ... + 19*x7*x9 + 40*x8*x9 + 49*x1
sage: f.degree()
3

We show that the default method does not sample uniformly at random from the ideal:

sage: P.<x,y,z> = GF(127)[]
sage: G = Sequence([x+7, y-2, z+110])
sage: I = Ideal([sum(P.random_element() * g for g in G) for _ in range(4)])
sage: all(I.random_element(degree=1) == 0 for _ in range(100))
True

If degree equals the degree of the generators a random linear combination of the generators is returned:

sage: P.<x,y> = QQ[]
sage: I = P.ideal([x^2,y^2])
sage: set_random_seed(5)
sage: I.random_element(degree=2)
-2*x^2 + 2*y^2

reduce(f)

Reduce an element modulo the reduced Groebner basis for this ideal. This returns 0 if and only if the element is in this ideal. In any case, this reduction is unique up to monomial orders.

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQ, 2)
sage: I = (x^3 + y, y)*R
sage: I.reduce(y)
0
sage: I.reduce(x^3)
0
sage: I.reduce(x - y)
x

sage: I = (y^2 - (x^3 + x))*R
sage: I.reduce(x^3)
y^2 - x
sage: I.reduce(x^6)
y^4 - 2*x*y^2 + x^2
sage: (y^2 - x)^2
y^4 - 2*x*y^2 + x^2

Note

Requires computation of a Groebner basis, which can be a very expensive operation.

subs(in_dict=None, **kwds)

Substitute variables.

This method substitutes some variables in the polynomials that generate the ideal with given values. Variables that are not specified in the input remain unchanged.

INPUT:

• in_dict – (optional) dictionary of inputs

• **kwds – named parameters

OUTPUT:

A new ideal with modified generators. If possible, in the same polynomial ring. Raises a TypeError if no common polynomial ring of the substituted generators can be found.

EXAMPLES:

sage: R.<x,y> = PolynomialRing(ZZ,2,'xy')
sage: I = R.ideal(x^5+y^5, x^2 + y + x^2*y^2 + 5); I
Ideal (x^5 + y^5, x^2*y^2 + x^2 + y + 5) of Multivariate Polynomial Ring in x, y over Integer Ring
sage: I.subs(x=y)
Ideal (2*y^5, y^4 + y^2 + y + 5) of Multivariate Polynomial Ring in x, y over Integer Ring
sage: I.subs({x:y})    # same substitution but with dictionary
Ideal (2*y^5, y^4 + y^2 + y + 5) of Multivariate Polynomial Ring in x, y over Integer Ring

The new ideal can be in a different ring:

sage: R.<a,b> = PolynomialRing(QQ,2)
sage: S.<x,y> = PolynomialRing(QQ,2)
sage: I = R.ideal(a^2+b^2+a-b+2); I
Ideal (a^2 + b^2 + a - b + 2) of Multivariate Polynomial Ring in a, b over Rational Field
sage: I.subs(a=x, b=y)
Ideal (x^2 + y^2 + x - y + 2) of Multivariate Polynomial Ring in x, y over Rational Field

The resulting ring need not be a multivariate polynomial ring:

sage: T.<t> = PolynomialRing(QQ)
sage: I.subs(a=t, b=t)
Principal ideal (t^2 + 1) of Univariate Polynomial Ring in t over Rational Field
sage: var("z")
z
sage: I.subs(a=z, b=z)
Principal ideal (2*z^2 + 2) of Symbolic Ring

Variables that are not substituted remain unchanged:

sage: R.<x,y> = PolynomialRing(QQ,2)
sage: I = R.ideal(x^2+y^2+x-y+2); I
Ideal (x^2 + y^2 + x - y + 2) of Multivariate Polynomial Ring in x, y over Rational Field
sage: I.subs(x=1)
Ideal (y^2 - y + 4) of Multivariate Polynomial Ring in x, y over Rational Field

weil_restriction()

Compute the Weil restriction of this ideal over some extension field. If the field is a finite field, then this computes the Weil restriction to the prime subfield.

A Weil restriction of scalars - denoted \(Res_{L/k}\) - is a functor which, for any finite extension of fields \(L/k\) and any algebraic variety \(X\) over \(L\), produces another corresponding variety \(Res_{L/k}(X)\), defined over \(k\). It is useful for reducing questions about varieties over large fields to questions about more complicated varieties over smaller fields.

This function does not compute this Weil restriction directly but computes on generating sets of polynomial ideals:

Let \(d\) be the degree of the field extension \(L/k\), let \(a\) a generator of \(L/k\) and \(p\) the minimal polynomial of \(L/k\). Denote this ideal by \(I\).

Specifically, this function first maps each variable \(x\) to its representation over \(k\): \(\sum_{i=0}^{d-1} a^i x_i\). Then each generator of \(I\) is evaluated over these representations and reduced modulo the minimal polynomial \(p\). The result is interpreted as a univariate polynomial in \(a\) and its coefficients are the new generators of the returned ideal.

If the input and the output ideals are radical, this is equivalent to the statement about algebraic varieties above.

OUTPUT: MPolynomial Ideal

EXAMPLES:

sage: k.<a> = GF(2^2)
sage: P.<x,y> = PolynomialRing(k,2)
sage: I = Ideal([x*y + 1, a*x + 1])
sage: I.variety()
[{y: a, x: a + 1}]
sage: J = I.weil_restriction()
sage: J
Ideal (x0*y0 + x1*y1 + 1, x1*y0 + x0*y1 + x1*y1, x1 + 1, x0 + x1) of
Multivariate Polynomial Ring in x0, x1, y0, y1 over Finite Field of size
2
sage: J += sage.rings.ideal.FieldIdeal(J.ring()) # ensure radical ideal
sage: J.variety() # py2
[{y1: 1, x1: 1, x0: 1, y0: 0}]
sage: J.variety() # py3
[{y1: 1, y0: 0, x1: 1, x0: 1}]

sage: J.weil_restriction() # returns J
Ideal (x0*y0 + x1*y1 + 1, x1*y0 + x0*y1 + x1*y1, x1 + 1, x0 + x1, x0^2 +
x0, x1^2 + x1, y0^2 + y0, y1^2 + y1) of Multivariate Polynomial Ring in
x0, x1, y0, y1 over Finite Field of size 2

sage: k.<a> = GF(3^5)
sage: P.<x,y,z> = PolynomialRing(k)
sage: I = sage.rings.ideal.Katsura(P)
sage: I.dimension()
0
sage: I.variety() # py2
[{y: 0, z: 0, x: 1}]
sage: I.variety() # py3
 [{z: 0, y: 0, x: 1}]

sage: J = I.weil_restriction(); J
Ideal (x0 - y0 - z0 - 1, x1 - y1 - z1, x2 - y2 - z2, x3 - y3 - z3, x4 -
y4 - z4, x0^2 + x2*x3 + x1*x4 - y0^2 - y2*y3 - y1*y4 - z0^2 - z2*z3 -
z1*z4 - x0, -x0*x1 - x2*x3 - x3^2 - x1*x4 + x2*x4 + y0*y1 + y2*y3 + y3^2
+ y1*y4 - y2*y4 + z0*z1 + z2*z3 + z3^2 + z1*z4 - z2*z4 - x1, x1^2 -
x0*x2 + x3^2 - x2*x4 + x3*x4 - y1^2 + y0*y2 - y3^2 + y2*y4 - y3*y4 -
z1^2 + z0*z2 - z3^2 + z2*z4 - z3*z4 - x2, -x1*x2 - x0*x3 - x3*x4 - x4^2
+ y1*y2 + y0*y3 + y3*y4 + y4^2 + z1*z2 + z0*z3 + z3*z4 + z4^2 - x3, x2^2
- x1*x3 - x0*x4 + x4^2 - y2^2 + y1*y3 + y0*y4 - y4^2 - z2^2 + z1*z3 +
z0*z4 - z4^2 - x4, -x0*y0 + x4*y1 + x3*y2 + x2*y3 + x1*y4 - y0*z0 +
y4*z1 + y3*z2 + y2*z3 + y1*z4 - y0, -x1*y0 - x0*y1 - x4*y1 - x3*y2 +
x4*y2 - x2*y3 + x3*y3 - x1*y4 + x2*y4 - y1*z0 - y0*z1 - y4*z1 - y3*z2 +
y4*z2 - y2*z3 + y3*z3 - y1*z4 + y2*z4 - y1, -x2*y0 - x1*y1 - x0*y2 -
x4*y2 - x3*y3 + x4*y3 - x2*y4 + x3*y4 - y2*z0 - y1*z1 - y0*z2 - y4*z2 -
y3*z3 + y4*z3 - y2*z4 + y3*z4 - y2, -x3*y0 - x2*y1 - x1*y2 - x0*y3 -
x4*y3 - x3*y4 + x4*y4 - y3*z0 - y2*z1 - y1*z2 - y0*z3 - y4*z3 - y3*z4 +
y4*z4 - y3, -x4*y0 - x3*y1 - x2*y2 - x1*y3 - x0*y4 - x4*y4 - y4*z0 -
y3*z1 - y2*z2 - y1*z3 - y0*z4 - y4*z4 - y4) of Multivariate Polynomial
Ring in x0, x1, x2, x3, x4, y0, y1, y2, y3, y4, z0, z1, z2, z3, z4 over
Finite Field of size 3
sage: J += sage.rings.ideal.FieldIdeal(J.ring()) # ensure radical ideal
sage: from sage.doctest.fixtures import reproducible_repr
sage: print(reproducible_repr(J.variety()))
[{x0: 1, x1: 0, x2: 0, x3: 0, x4: 0, y0: 0, y1: 0, y2: 0, y3: 0, y4: 0, z0: 0, z1: 0, z2: 0, z3: 0, z4: 0}]

Weil restrictions are often used to study elliptic curves over extension fields so we give a simple example involving those:

sage: K.<a> = QuadraticField(1/3)
sage: E = EllipticCurve(K,[1,2,3,4,5])

We pick a point on E:

sage: p = E.lift_x(1); p
(1 : 2 : 1)

sage: I = E.defining_ideal(); I
Ideal (-x^3 - 2*x^2*z + x*y*z + y^2*z - 4*x*z^2 + 3*y*z^2 - 5*z^3) of Multivariate Polynomial Ring in x, y, z over Number Field in a with defining polynomial x^2 - 1/3 with a = 0.5773502691896258?

Of course, the point p is a root of all generators of I:

sage: I.subs(x=1,y=2,z=1)
Ideal (0) of Multivariate Polynomial Ring in x, y, z over Number Field in a with defining polynomial x^2 - 1/3 with a = 0.5773502691896258?

I is also radical:

sage: I.radical() == I
True

So we compute its Weil restriction:

sage: J = I.weil_restriction()
sage: J
Ideal (-x0^3 - x0*x1^2 - 2*x0^2*z0 - 2/3*x1^2*z0 + x0*y0*z0 + y0^2*z0 +
1/3*x1*y1*z0 + 1/3*y1^2*z0 - 4*x0*z0^2 + 3*y0*z0^2 - 5*z0^3 -
4/3*x0*x1*z1 + 1/3*x1*y0*z1 + 1/3*x0*y1*z1 + 2/3*y0*y1*z1 - 8/3*x1*z0*z1
+ 2*y1*z0*z1 - 4/3*x0*z1^2 + y0*z1^2 - 5*z0*z1^2, -3*x0^2*x1 - 1/3*x1^3
- 4*x0*x1*z0 + x1*y0*z0 + x0*y1*z0 + 2*y0*y1*z0 - 4*x1*z0^2 + 3*y1*z0^2
- 2*x0^2*z1 - 2/3*x1^2*z1 + x0*y0*z1 + y0^2*z1 + 1/3*x1*y1*z1 +
1/3*y1^2*z1 - 8*x0*z0*z1 + 6*y0*z0*z1 - 15*z0^2*z1 - 4/3*x1*z1^2 +
y1*z1^2 - 5/3*z1^3) of Multivariate Polynomial Ring in x0, x1, y0, y1,
z0, z1 over Rational Field

We can check that the point p is still a root of all generators of J:

sage: J.subs(x0=1,y0=2,z0=1,x1=0,y1=0,z1=0)
Ideal (0, 0) of Multivariate Polynomial Ring in x0, x1, y0, y1, z0, z1 over Rational Field

Example for relative number fields:

sage: R.<x> = QQ[]
sage: K.<w> = NumberField(x^5-2)
sage: R.<x> = K[]
sage: L.<v> = K.extension(x^2+1)
sage: S.<x,y> = L[]
sage: I = S.ideal([y^2-x^3-1])
sage: I.weil_restriction()
Ideal (-x0^3 + 3*x0*x1^2 + y0^2 - y1^2 - 1, -3*x0^2*x1 + x1^3 + 2*y0*y1)
of Multivariate Polynomial Ring in x0, x1, y0, y1 over Number Field in w
with defining polynomial x^5 - 2

Note

Based on a Singular implementation by Michael Brickenstein

class sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal_macaulay2_repr

Bases: object

An ideal in a multivariate polynomial ring, which has an underlying Macaulay2 ring associated to it.

EXAMPLES:

sage: R.<x,y,z,w> = PolynomialRing(ZZ, 4)
sage: I = ideal(x*y-z^2, y^2-w^2)
sage: I
Ideal (x*y - z^2, y^2 - w^2) of Multivariate Polynomial Ring in x, y, z, w over Integer Ring

class sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal_magma_repr

Bases: object

class sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal_singular_base_repr

Bases: object

syzygy_module()

Computes the first syzygy (i.e., the module of relations of the given generators) of the ideal.

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQ)
sage: f = 2*x^2 + y
sage: g = y
sage: h = 2*f + g
sage: I = Ideal([f,g,h])
sage: M = I.syzygy_module(); M
[       -2        -1         1]
[       -y 2*x^2 + y         0]
sage: G = vector(I.gens())
sage: M*G
(0, 0)

ALGORITHM: Uses Singular’s syz command

class sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal_singular_repr

Bases: sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal_singular_base_repr

An ideal in a multivariate polynomial ring, which has an underlying Singular ring associated to it.

associated_primes(algorithm='sy')

Return a list of the associated primes of primary ideals of which the intersection is \(I\) = self.

An ideal \(Q\) is called primary if it is a proper ideal of the ring \(R\) and if whenever \(ab \in Q\) and \(a \not\in Q\) then \(b^n \in Q\) for some \(n \in \ZZ\).

If \(Q\) is a primary ideal of the ring \(R\), then the radical ideal \(P\) of \(Q\), i.e. \(P = \{a \in R, a^n \in Q\}\) for some \(n \in \ZZ\), is called the associated prime of \(Q\).

If \(I\) is a proper ideal of the ring \(R\) then there exists a decomposition in primary ideals \(Q_i\) such that

• their intersection is \(I\)

• none of the \(Q_i\) contains the intersection of the rest, and

• the associated prime ideals of \(Q_i\) are pairwise different.

This method returns the associated primes of the \(Q_i\).

INPUT:

• algorithm - string:

• 'sy' - (default) use the Shimoyama-Yokoyama algorithm

• 'gtz' - use the Gianni-Trager-Zacharias algorithm

OUTPUT:

• list - a list of associated primes

EXAMPLES:

sage: R.<x,y,z> = PolynomialRing(QQ, 3, order='lex')
sage: p = z^2 + 1; q = z^3 + 2
sage: I = (p*q^2, y-z^2)*R
sage: pd = I.associated_primes(); sorted(pd, key=str)
[Ideal (z^2 + 1, y + 1) of Multivariate Polynomial Ring in x, y, z over Rational Field,
 Ideal (z^3 + 2, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field]

ALGORITHM:

Uses Singular.

REFERENCES:

• Thomas Becker and Volker Weispfenning. Groebner Bases - A Computational Approach To Commutative Algebra. Springer, New York 1993.

basis_is_groebner(singular=Singular)

Return True if the generators of this ideal (self.gens()) form a Groebner basis.

Let \(I\) be the set of generators of this ideal. The check is performed by trying to lift \(Syz(LM(I))\) to \(Syz(I)\) as \(I\) forms a Groebner basis if and only if for every element \(S\) in \(Syz(LM(I))\):

\[S * G = \sum_{i=0}^{m} h_ig_i ---->_G 0.\]

ALGORITHM:

Uses Singular.

EXAMPLES:

sage: R.<a,b,c,d,e,f,g,h,i,j> = PolynomialRing(GF(127),10)
sage: I = sage.rings.ideal.Cyclic(R,4)
sage: I.basis_is_groebner()
False
sage: I2 = Ideal(I.groebner_basis())
sage: I2.basis_is_groebner()
True

A more complicated example:

sage: R.<U6,U5,U4,U3,U2, u6,u5,u4,u3,u2, h> = PolynomialRing(GF(7583))
sage: l = [u6 + u5 + u4 + u3 + u2 - 3791*h, \
           U6 + U5 + U4 + U3 + U2 - 3791*h, \
           U2*u2 - h^2, U3*u3 - h^2, U4*u4 - h^2, \
           U5*u4 + U5*u3 + U4*u3 + U5*u2 + U4*u2 + U3*u2 - 3791*U5*h - 3791*U4*h - 3791*U3*h - 3791*U2*h - 2842*h^2, \
           U4*u5 + U3*u5 + U2*u5 + U3*u4 + U2*u4 + U2*u3 - 3791*u5*h - 3791*u4*h - 3791*u3*h - 3791*u2*h - 2842*h^2, \
           U5*u5 - h^2, U4*U2*u3 + U5*U3*u2 + U4*U3*u2 + U3^2*u2 - 3791*U5*U3*h - 3791*U4*U3*h - 3791*U3^2*h - 3791*U5*U2*h \
            - 3791*U4*U2*h + U3*U2*h - 3791*U2^2*h - 3791*U4*u3*h - 3791*U4*u2*h - 3791*U3*u2*h - 2843*U5*h^2 + 1897*U4*h^2 - 946*U3*h^2 - 947*U2*h^2 + 2370*h^3, \
           U3*u5*u4 + U2*u5*u4 + U3*u4^2 + U2*u4^2 + U2*u4*u3 - 3791*u5*u4*h - 3791*u4^2*h - 3791*u4*u3*h - 3791*u4*u2*h + u5*h^2 - 2842*u4*h^2, \
           U2*u5*u4*u3 + U2*u4^2*u3 + U2*u4*u3^2 - 3791*u5*u4*u3*h - 3791*u4^2*u3*h - 3791*u4*u3^2*h - 3791*u4*u3*u2*h + u5*u4*h^2 + u4^2*h^2 + u5*u3*h^2 - 2842*u4*u3*h^2, \
           U5^2*U4*u3 + U5*U4^2*u3 + U5^2*U4*u2 + U5*U4^2*u2 + U5^2*U3*u2 + 2*U5*U4*U3*u2 + U5*U3^2*u2 - 3791*U5^2*U4*h - 3791*U5*U4^2*h - 3791*U5^2*U3*h \
            + U5*U4*U3*h - 3791*U5*U3^2*h - 3791*U5^2*U2*h + U5*U4*U2*h + U5*U3*U2*h - 3791*U5*U2^2*h - 3791*U5*U3*u2*h - 2842*U5^2*h^2 + 1897*U5*U4*h^2 \
            - U4^2*h^2 - 947*U5*U3*h^2 - U4*U3*h^2 - 948*U5*U2*h^2 - U4*U2*h^2 - 1422*U5*h^3 + 3791*U4*h^3, \
           u5*u4*u3*u2*h + u4^2*u3*u2*h + u4*u3^2*u2*h + u4*u3*u2^2*h + 2*u5*u4*u3*h^2 + 2*u4^2*u3*h^2 + 2*u4*u3^2*h^2 + 2*u5*u4*u2*h^2 + 2*u4^2*u2*h^2 \
            + 2*u5*u3*u2*h^2 + 1899*u4*u3*u2*h^2, \
           U5^2*U4*U3*u2 + U5*U4^2*U3*u2 + U5*U4*U3^2*u2 - 3791*U5^2*U4*U3*h - 3791*U5*U4^2*U3*h - 3791*U5*U4*U3^2*h - 3791*U5*U4*U3*U2*h \
            + 3791*U5*U4*U3*u2*h + U5^2*U4*h^2 + U5*U4^2*h^2 + U5^2*U3*h^2 - U4^2*U3*h^2 - U5*U3^2*h^2 - U4*U3^2*h^2 - U5*U4*U2*h^2 \
            - U5*U3*U2*h^2 - U4*U3*U2*h^2 + 3791*U5*U4*h^3 + 3791*U5*U3*h^3 + 3791*U4*U3*h^3, \
           u4^2*u3*u2*h^2 + 1515*u5*u3^2*u2*h^2 + u4*u3^2*u2*h^2 + 1515*u5*u4*u2^2*h^2 + 1515*u5*u3*u2^2*h^2 + u4*u3*u2^2*h^2 \
            + 1521*u5*u4*u3*h^3 - 3028*u4^2*u3*h^3 - 3028*u4*u3^2*h^3 + 1521*u5*u4*u2*h^3 - 3028*u4^2*u2*h^3 + 1521*u5*u3*u2*h^3 + 3420*u4*u3*u2*h^3, \
           U5^2*U4*U3*U2*h + U5*U4^2*U3*U2*h + U5*U4*U3^2*U2*h + U5*U4*U3*U2^2*h + 2*U5^2*U4*U3*h^2 + 2*U5*U4^2*U3*h^2 + 2*U5*U4*U3^2*h^2 \
            + 2*U5^2*U4*U2*h^2 + 2*U5*U4^2*U2*h^2 + 2*U5^2*U3*U2*h^2 - 2*U4^2*U3*U2*h^2 - 2*U5*U3^2*U2*h^2 - 2*U4*U3^2*U2*h^2 \
             - 2*U5*U4*U2^2*h^2 - 2*U5*U3*U2^2*h^2 - 2*U4*U3*U2^2*h^2 - U5*U4*U3*h^3 - U5*U4*U2*h^3 - U5*U3*U2*h^3 - U4*U3*U2*h^3]

sage: Ideal(l).basis_is_groebner()
False
sage: gb = Ideal(l).groebner_basis()
sage: Ideal(gb).basis_is_groebner()
True

Note

From the Singular Manual for the reduce function we use in this method: ‘The result may have no meaning if the second argument (self) is not a standard basis’. I (malb) believe this refers to the mathematical fact that the results may have no meaning if self is no standard basis, i.e., Singular doesn’t ‘add’ any additional ‘nonsense’ to the result. So we may actually use reduce to determine if self is a Groebner basis.

complete_primary_decomposition(object)

A decorator that creates a cached version of an instance method of a class.

Note

For proper behavior, the method must be a pure function (no side effects). Arguments to the method must be hashable or transformed into something hashable using key or they must define sage.structure.sage_object.SageObject._cache_key().

EXAMPLES:

sage: class Foo(object):
....:     @cached_method
....:     def f(self, t, x=2):
....:         print('computing')
....:         return t**x
sage: a = Foo()

The example shows that the actual computation takes place only once, and that the result is identical for equivalent input:

sage: res = a.f(3, 2); res
computing
9
sage: a.f(t = 3, x = 2) is res
True
sage: a.f(3) is res
True

Note, however, that the CachedMethod is replaced by a CachedMethodCaller or CachedMethodCallerNoArgs as soon as it is bound to an instance or class:

sage: P.<a,b,c,d> = QQ[]
sage: I = P*[a,b]
sage: type(I.__class__.gens)
<type 'sage.misc.cachefunc.CachedMethodCallerNoArgs'>

So, you would hardly ever see an instance of this class alive.

The parameter key can be used to pass a function which creates a custom cache key for inputs. In the following example, this parameter is used to ignore the algorithm keyword for caching:

sage: class A(object):
....:     def _f_normalize(self, x, algorithm): return x
....:     @cached_method(key=_f_normalize)
....:     def f(self, x, algorithm='default'): return x
sage: a = A()
sage: a.f(1, algorithm="default") is a.f(1) is a.f(1, algorithm="algorithm")
True

The parameter do_pickle can be used to enable pickling of the cache. Usually the cache is not stored when pickling:

sage: class A(object):
....:     @cached_method
....:     def f(self, x): return None
sage: import __main__
sage: __main__.A = A
sage: a = A()
sage: a.f(1)
sage: len(a.f.cache)
1
sage: b = loads(dumps(a))
sage: len(b.f.cache)
0

When do_pickle is set, the pickle contains the contents of the cache:

sage: class A(object):
....:     @cached_method(do_pickle=True)
....:     def f(self, x): return None
sage: __main__.A = A
sage: a = A()
sage: a.f(1)
sage: len(a.f.cache)
1
sage: b = loads(dumps(a))
sage: len(b.f.cache)
1

Cached methods cannot be copied like usual methods, see trac ticket #12603. Copying them can lead to very surprising results:

sage: class A:
....:     @cached_method
....:     def f(self):
....:         return 1
sage: class B:
....:     g=A.f
....:     def f(self):
....:         return 2

sage: b=B()
sage: b.f()
2
sage: b.g()
1
sage: b.f()
1

dimension(singular='singular_default')

The dimension of the ring modulo this ideal.

EXAMPLES:

sage: P.<x,y,z> = PolynomialRing(GF(32003),order='degrevlex')
sage: I = ideal(x^2-y,x^3)
sage: I.dimension()
1

If the ideal is the total ring, the dimension is \(-1\) by convention.

For polynomials over a finite field of order too large for Singular, this falls back on a toy implementation of Buchberger to compute the Groebner basis, then uses the algorithm described in Chapter 9, Section 1 of Cox, Little, and O’Shea’s “Ideals, Varieties, and Algorithms”.

EXAMPLES:

sage: R.<x,y> = PolynomialRing(GF(2147483659),order='lex')
sage: I = R.ideal([x*y,x*y+1])
sage: I.dimension()
verbose 0 (...: multi_polynomial_ideal.py, dimension) Warning: falling back to very slow toy implementation.
-1
sage: I=ideal([x*(x*y+1),y*(x*y+1)])
sage: I.dimension()
verbose 0 (...: multi_polynomial_ideal.py, dimension) Warning: falling back to very slow toy implementation.
1
sage: I = R.ideal([x^3*y,x*y^2])
sage: I.dimension()
verbose 0 (...: multi_polynomial_ideal.py, dimension) Warning: falling back to very slow toy implementation.
1
sage: R.<x,y> = PolynomialRing(GF(2147483659),order='lex')
sage: I = R.ideal(0)
sage: I.dimension()
verbose 0 (...: multi_polynomial_ideal.py, dimension) Warning: falling back to very slow toy implementation.
2

ALGORITHM:

Uses Singular, unless the characteristic is too large.

Note

Requires computation of a Groebner basis, which can be a very expensive operation.

elimination_ideal(variables, algorithm=None, *args, **kwds)

Return the elimination ideal of this ideal with respect to the variables given in variables.

INPUT:

• variables – a list or tuple of variables in self.ring()

• algorithm - determines the algorithm to use, see below

  for available algorithms.

ALGORITHMS:

• 'libsingular:eliminate' – libSingular’s eliminate command (default)

• 'giac:eliminate' – Giac’s eliminate command (if available)

If only a system is given - e.g. ‘giac’ - the default algorithm is chosen for that system.

EXAMPLES:

sage: R.<x,y,t,s,z> = PolynomialRing(QQ,5)
sage: I = R * [x-t,y-t^2,z-t^3,s-x+y^3]
sage: J = I.elimination_ideal([t,s]); J
Ideal (y^2 - x*z, x*y - z, x^2 - y) of Multivariate
Polynomial Ring in x, y, t, s, z over Rational Field

You can use Giac to compute the elimination ideal:

sage: print("possible output from giac", flush=True); I.elimination_ideal([t, s], algorithm="giac") == J
possible output...
True

The list of available Giac options is provided at sage.libs.giac.groebner_basis().

ALGORITHM:

Uses Singular, or Giac (if available).

Note

Requires computation of a Groebner basis, which can be a very expensive operation.

genus(object)

A decorator that creates a cached version of an instance method of a class.

Note

For proper behavior, the method must be a pure function (no side effects). Arguments to the method must be hashable or transformed into something hashable using key or they must define sage.structure.sage_object.SageObject._cache_key().

EXAMPLES:

sage: class Foo(object):
....:     @cached_method
....:     def f(self, t, x=2):
....:         print('computing')
....:         return t**x
sage: a = Foo()

The example shows that the actual computation takes place only once, and that the result is identical for equivalent input:

sage: res = a.f(3, 2); res
computing
9
sage: a.f(t = 3, x = 2) is res
True
sage: a.f(3) is res
True

Note, however, that the CachedMethod is replaced by a CachedMethodCaller or CachedMethodCallerNoArgs as soon as it is bound to an instance or class:

sage: P.<a,b,c,d> = QQ[]
sage: I = P*[a,b]
sage: type(I.__class__.gens)
<type 'sage.misc.cachefunc.CachedMethodCallerNoArgs'>

So, you would hardly ever see an instance of this class alive.

The parameter key can be used to pass a function which creates a custom cache key for inputs. In the following example, this parameter is used to ignore the algorithm keyword for caching:

sage: class A(object):
....:     def _f_normalize(self, x, algorithm): return x
....:     @cached_method(key=_f_normalize)
....:     def f(self, x, algorithm='default'): return x
sage: a = A()
sage: a.f(1, algorithm="default") is a.f(1) is a.f(1, algorithm="algorithm")
True

The parameter do_pickle can be used to enable pickling of the cache. Usually the cache is not stored when pickling:

sage: class A(object):
....:     @cached_method
....:     def f(self, x): return None
sage: import __main__
sage: __main__.A = A
sage: a = A()
sage: a.f(1)
sage: len(a.f.cache)
1
sage: b = loads(dumps(a))
sage: len(b.f.cache)
0

When do_pickle is set, the pickle contains the contents of the cache:

sage: class A(object):
....:     @cached_method(do_pickle=True)
....:     def f(self, x): return None
sage: __main__.A = A
sage: a = A()
sage: a.f(1)
sage: len(a.f.cache)
1
sage: b = loads(dumps(a))
sage: len(b.f.cache)
1

Cached methods cannot be copied like usual methods, see trac ticket #12603. Copying them can lead to very surprising results:

sage: class A:
....:     @cached_method
....:     def f(self):
....:         return 1
sage: class B:
....:     g=A.f
....:     def f(self):
....:         return 2

sage: b=B()
sage: b.f()
2
sage: b.g()
1
sage: b.f()
1

hilbert_numerator(grading=None, algorithm='sage')

Return the Hilbert numerator of this ideal.

INPUT:

• grading – (optional) a list or tuple of integers

• algorithm – (default: 'sage') must be either 'sage' or 'singular'

Let \(I\) (which is self) be a homogeneous ideal and \(R = \bigoplus_d R_d\) (which is self.ring()) be a graded commutative algebra over a field \(K\). Then the Hilbert function is defined as \(H(d) = dim_K R_d\) and the Hilbert series of \(I\) is defined as the formal power series \(HS(t) = \sum_{d=0}^{\infty} H(d) t^d\).

This power series can be expressed as \(HS(t) = Q(t) / (1-t)^n\) where \(Q(t)\) is a polynomial over \(Z\) and \(n\) the number of variables in \(R\). This method returns \(Q(t)\), the numerator; hence the name, hilbert_numerator. An optional grading can be given, in which case the graded (or weighted) Hilbert numerator is given.

EXAMPLES:

sage: P.<x,y,z> = PolynomialRing(QQ)
sage: I = Ideal([x^3*y^2 + 3*x^2*y^2*z + y^3*z^2 + z^5])
sage: I.hilbert_numerator()
-t^5 + 1
sage: R.<a,b> = PolynomialRing(QQ)
sage: J = R.ideal([a^2*b,a*b^2])
sage: J.hilbert_numerator()
t^4 - 2*t^3 + 1
sage: J.hilbert_numerator(grading=(10,3))
t^26 - t^23 - t^16 + 1

hilbert_polynomial(algorithm='sage')

Return the Hilbert polynomial of this ideal.

INPUT:

• algorithm – (default: 'sage') must be either 'sage' or 'singular'

Let \(I\) (which is self) be a homogeneous ideal and \(R = \bigoplus_d R_d\) (which is self.ring()) be a graded commutative algebra over a field \(K\). The Hilbert polynomial is the unique polynomial \(HP(t)\) with rational coefficients such that \(HP(d) = dim_K R_d\) for all but finitely many positive integers \(d\).

EXAMPLES:

sage: P.<x,y,z> = PolynomialRing(QQ)
sage: I = Ideal([x^3*y^2 + 3*x^2*y^2*z + y^3*z^2 + z^5])
sage: I.hilbert_polynomial()
5*t - 5

Of course, the Hilbert polynomial of a zero-dimensional ideal is zero:

sage: J0 = Ideal([x^3*y^2 + 3*x^2*y^2*z + y^3*z^2 + z^5, y^3-2*x*z^2+x*y,x^4+x*y-y*z^2])
sage: J = P*[m.lm() for m in J0.groebner_basis()]
sage: J.dimension()
0
sage: J.hilbert_polynomial()
0

It is possible to request a computation using the Singular library:

sage: I.hilbert_polynomial(algorithm = 'singular') == I.hilbert_polynomial()
True
sage: J.hilbert_polynomial(algorithm = 'singular') == J.hilbert_polynomial()
True

Here is a bigger examples:

sage: n = 4; m = 11; P = PolynomialRing(QQ, n * m, "x"); x = P.gens(); M = Matrix(n, x)
sage: Minors = P.ideal(M.minors(2))
sage: hp = Minors.hilbert_polynomial(); hp
1/21772800*t^13 + 61/21772800*t^12 + 1661/21772800*t^11
 + 26681/21772800*t^10 + 93841/7257600*t^9 + 685421/7257600*t^8
 + 1524809/3110400*t^7 + 39780323/21772800*t^6 + 6638071/1360800*t^5
 + 12509761/1360800*t^4 + 2689031/226800*t^3 + 1494509/151200*t^2
 + 12001/2520*t + 1

Because Singular uses 32-bit integers, the above example would fail with Singular. We don’t test it here, as it has a side-effect on other tests that is not understood yet (see trac ticket #26300):

sage: Minors.hilbert_polynomial(algorithm = 'singular')    # not tested
Traceback (most recent call last):
...
RuntimeError: error in Singular function call 'hilbPoly':
int overflow in hilb 1
error occurred in or before poly.lib::hilbPoly line 58: `   intvec v=hilb(I,2);`
expected intvec-expression. type 'help intvec;'

Note that in this example, the Hilbert polynomial gives the coefficients of the Hilbert-Poincaré series in all degrees:

sage: P = PowerSeriesRing(QQ, 't', default_prec = 50)
sage: hs = Minors.hilbert_series()
sage: list(P(hs.numerator()) / P(hs.denominator())) == [hp(t = k) for k in range(50)]
True

hilbert_series(grading=None, algorithm='sage')

Return the Hilbert series of this ideal.

INPUT:

• grading – (optional) a list or tuple of integers

• algorithm – (default: 'sage') must be either 'sage' or 'singular'

Let \(I\) (which is self) be a homogeneous ideal and \(R = \bigoplus_d R_d\) (which is self.ring()) be a graded commutative algebra over a field \(K\). Then the Hilbert function is defined as \(H(d) = dim_K R_d\) and the Hilbert series of \(I\) is defined as the formal power series \(HS(t) = \sum_{d=0}^{\infty} H(d) t^d\).

This power series can be expressed as \(HS(t) = Q(t) / (1-t)^n\) where \(Q(t)\) is a polynomial over \(Z\) and \(n\) the number of variables in \(R\). This method returns \(Q(t) / (1-t)^n\), normalised so that the leading monomial of the numerator is positive.

An optional grading can be given, in which case the graded (or weighted) Hilbert series is given.

EXAMPLES:

sage: P.<x,y,z> = PolynomialRing(QQ)
sage: I = Ideal([x^3*y^2 + 3*x^2*y^2*z + y^3*z^2 + z^5])
sage: I.hilbert_series()
(t^4 + t^3 + t^2 + t + 1)/(t^2 - 2*t + 1)
sage: R.<a,b> = PolynomialRing(QQ)
sage: J = R.ideal([a^2*b,a*b^2])
sage: J.hilbert_series()
(t^3 - t^2 - t - 1)/(t - 1)
sage: J.hilbert_series(grading=(10,3))
(t^25 + t^24 + t^23 - t^15 - t^14 - t^13 - t^12 - t^11
 - t^10 - t^9 - t^8 - t^7 - t^6 - t^5 - t^4 - t^3 - t^2
 - t - 1)/(t^12 + t^11 + t^10 - t^2 - t - 1)

sage: K = R.ideal([a^2*b^3, a*b^4 + a^3*b^2])
sage: K.hilbert_series(grading=[1,2])
(t^11 + t^8 - t^6 - t^5 - t^4 - t^3 - t^2 - t - 1)/(t^2 - 1)
sage: K.hilbert_series(grading=[2,1])
(2*t^7 - t^6 - t^4 - t^2 - 1)/(t - 1)

integral_closure(p=0, r=True, singular='singular_default')

Let \(I\) = self.

Return the integral closure of \(I, ..., I^p\), where \(sI\) is an ideal in the polynomial ring \(R=k[x(1),...x(n)]\). If \(p\) is not given, or \(p=0\), compute the closure of all powers up to the maximum degree in t occurring in the closure of \(R[It]\) (so this is the last power whose closure is not just the sum/product of the smaller). If \(r\) is given and r is True, I.integral_closure() starts with a check whether I is already a radical ideal.

INPUT:

• p - powers of I (default: 0)

• r - check whether self is a radical ideal first (default: True)

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: I = ideal([x^2,x*y^4,y^5])
sage: I.integral_closure()
[x^2, x*y^4, y^5, x*y^3]

ALGORITHM:

Uses libSINGULAR.

interreduced_basis()

If this ideal is spanned by \((f_1, ..., f_n)\) this method returns \((g_1, ..., g_s)\) such that:

• \((f_1,...,f_n) = (g_1,...,g_s)\)

• \(LT(g_i) != LT(g_j)\) for all \(i != j\)

• \(LT(g_i)\) does not divide \(m\) for all monomials \(m\) of

  \(\{g_1,...,g_{i-1},g_{i+1},...,g_s\}\)

• \(LC(g_i) == 1\) for all \(i\) if the coefficient ring is a field.

EXAMPLES:

sage: R.<x,y,z> = PolynomialRing(QQ)
sage: I = Ideal([z*x+y^3,z+y^3,z+x*y])
sage: I.interreduced_basis()
[y^3 + z, x*y + z, x*z - z]

Note that tail reduction for local orderings is not well-defined:

sage: R.<x,y,z> = PolynomialRing(QQ,order='negdegrevlex')
sage: I = Ideal([z*x+y^3,z+y^3,z+x*y])
sage: I.interreduced_basis()
[z + x*y, x*y - y^3, x^2*y - y^3]

A fixed error with nonstandard base fields:

sage: R.<t>=QQ['t']
sage: K.<x,y>=R.fraction_field()['x,y']
sage: I=t*x*K
sage: I.interreduced_basis()
[x]

The interreduced basis of 0 is 0:

sage: P.<x,y,z> = GF(2)[]
sage: Ideal(P(0)).interreduced_basis()
[0]

ALGORITHM:

Uses Singular’s interred command or sage.rings.polynomial.toy_buchberger.inter_reduction() if conversion to Singular fails.

intersection(*others)

Return the intersection of the arguments with this ideal.

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQ, 2, order='lex')
sage: I = x*R
sage: J = y*R
sage: I.intersection(J)
Ideal (x*y) of Multivariate Polynomial Ring in x, y over Rational Field

The following simple example illustrates that the product need not equal the intersection.

sage: I = (x^2, y)*R
sage: J = (y^2, x)*R
sage: K = I.intersection(J); K
Ideal (y^2, x*y, x^2) of Multivariate Polynomial Ring in x, y over Rational Field
sage: IJ = I*J; IJ
Ideal (x^2*y^2, x^3, y^3, x*y) of Multivariate Polynomial Ring in x, y over Rational Field
sage: IJ == K
False

Intersection of several ideals:

sage: R.<x,y,z> = PolynomialRing(QQ, 3, order='lex')
sage: I1 = x*R
sage: I2 = y*R
sage: I3 = (x, y)*R
sage: I4 = (x^2 + x*y*z, y^2 - z^3*y, z^3 + y^5*x*z)*R
sage: I1.intersection(I2, I3, I4).groebner_basis()
[x^2*y + x*y*z^4, x*y^2 - x*y*z^3, x*y*z^20 - x*y*z^3]

The ideals must share the same ring:

sage: R2.<x,y> = PolynomialRing(QQ, 2, order='lex')
sage: R3.<x,y,z> = PolynomialRing(QQ, 3, order='lex')
sage: I2 = x*R2
sage: I3 = x*R3
sage: I2.intersection(I3)
Traceback (most recent call last):
...
TypeError: Intersection is only available for ideals of the same ring.

is_prime(**kwds)

Return True if this ideal is prime.

INPUT:

• keyword arguments are passed on to complete_primary_decomposition; in this way you can specify the algorithm to use.

EXAMPLES:

sage: R.<x, y> = PolynomialRing(QQ, 2)
sage: I = (x^2 - y^2 - 1)*R
sage: I.is_prime()
True
sage: (I^2).is_prime()
False

sage: J = (x^2 - y^2)*R
sage: J.is_prime()
False
sage: (J^3).is_prime()
False

sage: (I * J).is_prime()
False

The following is trac ticket #5982. Note that the quotient ring is not recognized as being a field at this time, so the fraction field is not the quotient ring itself:

sage: Q = R.quotient(I); Q
Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 - y^2 - 1)
sage: Q.fraction_field()
Fraction Field of Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 - y^2 - 1)

minimal_associated_primes()

OUTPUT:

• list - a list of prime ideals

EXAMPLES:

sage: R.<x,y,z> = PolynomialRing(QQ, 3, 'xyz')
sage: p = z^2 + 1; q = z^3 + 2
sage: I = (p*q^2, y-z^2)*R
sage: sorted(I.minimal_associated_primes(), key=str)
[Ideal (z^2 + 1, -z^2 + y) of Multivariate Polynomial Ring in x, y, z over Rational Field,
 Ideal (z^3 + 2, -z^2 + y) of Multivariate Polynomial Ring in x, y, z over Rational Field]

ALGORITHM:

Uses Singular.

normal_basis(degree=None, algorithm='libsingular', singular='singular_default')

Return a vector space basis of the quotient ring of this ideal.

INPUT:

• degree – integer (default: None)

• algorithm – string (default: "libsingular"); if not the default, this will use the kbase() or weightKB() command from Singular

• singular – the singular interpreter to use when algorithm is not "libsingular" (default: the default instance)

OUTPUT:

Monomials in the basis. If degree is given, only the monomials of the given degree are returned.

EXAMPLES:

sage: R.<x,y,z> = PolynomialRing(QQ)
sage: I = R.ideal(x^2+y^2+z^2-4, x^2+2*y^2-5, x*z-1)
sage: I.normal_basis()
[y*z^2, z^2, y*z, z, x*y, y, x, 1]
sage: I.normal_basis(algorithm='singular')
[y*z^2, z^2, y*z, z, x*y, y, x, 1]

The result can be restricted to monomials of a chosen degree, which is particularly useful when the quotient ring is not finite-dimensional as a vector space.

sage: J = R.ideal(x^2+y^2+z^2-4, x^2+2*y^2-5)
sage: J.dimension()
1
sage: [J.normal_basis(d) for d in (0..3)]
[[1], [z, y, x], [z^2, y*z, x*z, x*y], [z^3, y*z^2, x*z^2, x*y*z]]
sage: [J.normal_basis(d, algorithm='singular') for d in (0..3)]
[[1], [z, y, x], [z^2, y*z, x*z, x*y], [z^3, y*z^2, x*z^2, x*y*z]]

In case of a polynomial ring with a weighted term order, the degree of the monomials is taken with respect to the weights.

sage: T = TermOrder('wdegrevlex', (1, 2, 3))
sage: R.<x,y,z> = PolynomialRing(QQ, order=T)
sage: B = R.ideal(x*y^2 + x^5, z*y + x^3*y).normal_basis(9); B
[x^2*y^2*z, x^3*z^2, x*y*z^2, z^3]
sage: all(f.degree() == 9 for f in B)
True

plot(singular=Singular)

If you somehow manage to install surf, perhaps you can use this function to implicitly plot the real zero locus of this ideal (if principal).

INPUT:

• self - must be a principal ideal in 2 or 3 vars over \(\QQ\).

EXAMPLES:

Implicit plotting in 2-d:

sage: R.<x,y> = PolynomialRing(QQ,2)
sage: I = R.ideal([y^3 - x^2])
sage: I.plot()        # cusp
Graphics object consisting of 1 graphics primitive
sage: I = R.ideal([y^2 - x^2 - 1])
sage: I.plot()        # hyperbola
Graphics object consisting of 1 graphics primitive
sage: I = R.ideal([y^2 + x^2*(1/4) - 1])
sage: I.plot()        # ellipse
Graphics object consisting of 1 graphics primitive
sage: I = R.ideal([y^2-(x^2-1)*(x-2)])
sage: I.plot()        # elliptic curve
Graphics object consisting of 1 graphics primitive

Implicit plotting in 3-d:

sage: R.<x,y,z> = PolynomialRing(QQ,3)
sage: I = R.ideal([y^2 + x^2*(1/4) - z])
sage: I.plot()          # a cone; optional - surf
sage: I = R.ideal([y^2 + z^2*(1/4) - x])
sage: I.plot()          # same code, from a different angle; optional - surf
sage: I = R.ideal([x^2*y^2+x^2*z^2+y^2*z^2-16*x*y*z])
sage: I.plot()          # Steiner surface; optional - surf

AUTHORS:

• David Joyner (2006-02-12)

primary_decomposition(algorithm='sy')

Return a list of primary ideals such that their intersection is self.

An ideal \(Q\) is called primary if it is a proper ideal of the ring \(R\), and if whenever \(ab \in Q\) and \(a \not\in Q\), then \(b^n \in Q\) for some \(n \in \ZZ\).

If \(Q\) is a primary ideal of the ring \(R\), then the radical ideal \(P\) of \(Q\) (i.e. the ideal consisting of all \(a \in R\) with a^n in Q` for some \(n \in \ZZ\)), is called the associated prime of \(Q\).

If \(I\) is a proper ideal of a Noetherian ring \(R\), then there exists a finite collection of primary ideals \(Q_i\) such that the following hold:

• the intersection of the \(Q_i\) is \(I\);

• none of the \(Q_i\) contains the intersection of the others;

• the associated prime ideals of the \(Q_i\) are pairwise distinct.

INPUT:

• algorithm – string:

  • 'sy' – (default) use the Shimoyama-Yokoyama algorithm

  • 'gtz' – use the Gianni-Trager-Zacharias algorithm

OUTPUT:

• a list of primary ideals \(Q_i\) forming a primary decomposition of self.

EXAMPLES:

sage: R.<x,y,z> = PolynomialRing(QQ, 3, order='lex')
sage: p = z^2 + 1; q = z^3 + 2
sage: I = (p*q^2, y-z^2)*R
sage: pd = I.primary_decomposition(); sorted(pd, key=str)
[Ideal (z^2 + 1, y + 1) of Multivariate Polynomial Ring in x, y, z over Rational Field,
 Ideal (z^6 + 4*z^3 + 4, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field]

sage: from functools import reduce
sage: reduce(lambda Qi,Qj: Qi.intersection(Qj), pd) == I
True

ALGORITHM:

Uses Singular.

REFERENCES:

• Thomas Becker and Volker Weispfenning. Groebner Bases - A Computational Approach To Commutative Algebra. Springer, New York 1993.

primary_decomposition_complete(object)

A decorator that creates a cached version of an instance method of a class.

Note

For proper behavior, the method must be a pure function (no side effects). Arguments to the method must be hashable or transformed into something hashable using key or they must define sage.structure.sage_object.SageObject._cache_key().

EXAMPLES:

sage: class Foo(object):
....:     @cached_method
....:     def f(self, t, x=2):
....:         print('computing')
....:         return t**x
sage: a = Foo()

The example shows that the actual computation takes place only once, and that the result is identical for equivalent input:

sage: res = a.f(3, 2); res
computing
9
sage: a.f(t = 3, x = 2) is res
True
sage: a.f(3) is res
True

Note, however, that the CachedMethod is replaced by a CachedMethodCaller or CachedMethodCallerNoArgs as soon as it is bound to an instance or class:

sage: P.<a,b,c,d> = QQ[]
sage: I = P*[a,b]
sage: type(I.__class__.gens)
<type 'sage.misc.cachefunc.CachedMethodCallerNoArgs'>

So, you would hardly ever see an instance of this class alive.

The parameter key can be used to pass a function which creates a custom cache key for inputs. In the following example, this parameter is used to ignore the algorithm keyword for caching:

sage: class A(object):
....:     def _f_normalize(self, x, algorithm): return x
....:     @cached_method(key=_f_normalize)
....:     def f(self, x, algorithm='default'): return x
sage: a = A()
sage: a.f(1, algorithm="default") is a.f(1) is a.f(1, algorithm="algorithm")
True

The parameter do_pickle can be used to enable pickling of the cache. Usually the cache is not stored when pickling:

sage: class A(object):
....:     @cached_method
....:     def f(self, x): return None
sage: import __main__
sage: __main__.A = A
sage: a = A()
sage: a.f(1)
sage: len(a.f.cache)
1
sage: b = loads(dumps(a))
sage: len(b.f.cache)
0

When do_pickle is set, the pickle contains the contents of the cache:

sage: class A(object):
....:     @cached_method(do_pickle=True)
....:     def f(self, x): return None
sage: __main__.A = A
sage: a = A()
sage: a.f(1)
sage: len(a.f.cache)
1
sage: b = loads(dumps(a))
sage: len(b.f.cache)
1

Cached methods cannot be copied like usual methods, see trac ticket #12603. Copying them can lead to very surprising results:

sage: class A:
....:     @cached_method
....:     def f(self):
....:         return 1
sage: class B:
....:     g=A.f
....:     def f(self):
....:         return 2

sage: b=B()
sage: b.f()
2
sage: b.g()
1
sage: b.f()
1

quotient(J)

Given ideals \(I\) = self and \(J\) in the same polynomial ring \(P\), return the ideal quotient of \(I\) by \(J\) consisting of the polynomials \(a\) of \(P\) such that \(\{aJ \subset I\}\).

This is also referred to as the colon ideal (\(I\):\(J\)).

INPUT:

• J - multivariate polynomial ideal

EXAMPLES:

sage: R.<x,y,z> = PolynomialRing(GF(181),3)
sage: I = Ideal([x^2+x*y*z,y^2-z^3*y,z^3+y^5*x*z])
sage: J = Ideal([x])
sage: Q = I.quotient(J)
sage: y*z + x in I
False
sage: x in J
True
sage: x * (y*z + x) in I
True

radical()

The radical of this ideal.

EXAMPLES:

This is an obviously not radical ideal:

sage: R.<x,y,z> = PolynomialRing(QQ, 3)
sage: I = (x^2, y^3, (x*z)^4 + y^3 + 10*x^2)*R
sage: I.radical()
Ideal (y, x) of Multivariate Polynomial Ring in x, y, z over Rational Field

That the radical is correct is clear from the Groebner basis.

sage: I.groebner_basis()
[y^3, x^2]

This is the example from the Singular manual:

sage: p = z^2 + 1; q = z^3 + 2
sage: I = (p*q^2, y-z^2)*R
sage: I.radical()
Ideal (z^2 - y, y^2*z + y*z + 2*y + 2) of Multivariate Polynomial Ring in x, y, z over Rational Field

Note

From the Singular manual: A combination of the algorithms of Krick/Logar and Kemper is used. Works also in positive characteristic (Kempers algorithm).

sage: R.<x,y,z> = PolynomialRing(GF(37), 3)
sage: p = z^2 + 1; q = z^3 + 2
sage: I = (p*q^2, y - z^2)*R
sage: I.radical()
Ideal (z^2 - y, y^2*z + y*z + 2*y + 2) of Multivariate Polynomial Ring in x, y, z over Finite Field of size 37

saturation(other)

Return the saturation (and saturation exponent) of the ideal self with respect to the ideal other

INPUT:

• other – another ideal in the same ring

OUTPUT:

• a pair (ideal, integer)

EXAMPLES:

sage: R.<x, y, z> = QQ[]
sage: I = R.ideal(x^5*z^3, x*y*z, y*z^4)
sage: J = R.ideal(z)
sage: I.saturation(J)
(Ideal (y, x^5) of Multivariate Polynomial Ring in x, y, z over Rational Field, 4)

syzygy_module()

Computes the first syzygy (i.e., the module of relations of the given generators) of the ideal.

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQ)
sage: f = 2*x^2 + y
sage: g = y
sage: h = 2*f + g
sage: I = Ideal([f,g,h])
sage: M = I.syzygy_module(); M
[       -2        -1         1]
[       -y 2*x^2 + y         0]
sage: G = vector(I.gens())
sage: M*G
(0, 0)

ALGORITHM:

Uses Singular’s syz command.

transformed_basis(algorithm='gwalk', other_ring=None, singular='singular_default')

Return a lex or other_ring Groebner Basis for this ideal.

INPUT:

• algorithm - see below for options.

• other_ring - only valid for algorithm ‘fglm’, if provided conversion will be performed to this ring. Otherwise a lex Groebner basis will be returned.

ALGORITHMS:

• fglm - FGLM algorithm. The input ideal must be given with a reduced Groebner Basis of a zero-dimensional ideal

• gwalk - Groebner Walk algorithm (default)

• awalk1 - ‘first alternative’ algorithm

• awalk2 - ‘second alternative’ algorithm

• twalk - Tran algorithm

• fwalk - Fractal Walk algorithm

EXAMPLES:

sage: R.<x,y,z> = PolynomialRing(QQ,3)
sage: I = Ideal([y^3+x^2,x^2*y+x^2, x^3-x^2, z^4-x^2-y])
sage: I = Ideal(I.groebner_basis())
sage: S.<z,x,y> = PolynomialRing(QQ,3,order='lex')
sage: J = Ideal(I.transformed_basis('fglm',S))
sage: J
Ideal (z^4 + y^3 - y, x^2 + y^3, x*y^3 - y^3, y^4 + y^3)
of Multivariate Polynomial Ring in z, x, y over Rational Field

sage: R.<z,y,x>=PolynomialRing(GF(32003),3,order='lex')
sage: I=Ideal([y^3+x*y*z+y^2*z+x*z^3,3+x*y+x^2*y+y^2*z])
sage: I.transformed_basis('gwalk')
[z*y^2 + y*x^2 + y*x + 3,
 z*x + 8297*y^8*x^2 + 8297*y^8*x + 3556*y^7 - 8297*y^6*x^4 + 15409*y^6*x^3 - 8297*y^6*x^2
 - 8297*y^5*x^5 + 15409*y^5*x^4 - 8297*y^5*x^3 + 3556*y^5*x^2 + 3556*y^5*x + 3556*y^4*x^3
 + 3556*y^4*x^2 - 10668*y^4 - 10668*y^3*x - 8297*y^2*x^9 - 1185*y^2*x^8 + 14224*y^2*x^7
 - 1185*y^2*x^6 - 8297*y^2*x^5 - 14223*y*x^7 - 10666*y*x^6 - 10666*y*x^5 - 14223*y*x^4
 + x^5 + 2*x^4 + x^3,
 y^9 - y^7*x^2 - y^7*x - y^6*x^3 - y^6*x^2 - 3*y^6 - 3*y^5*x - y^3*x^7 - 3*y^3*x^6
 - 3*y^3*x^5 - y^3*x^4 - 9*y^2*x^5 - 18*y^2*x^4 - 9*y^2*x^3 - 27*y*x^3 - 27*y*x^2 - 27*x]

ALGORITHM:

Uses Singular.

triangular_decomposition(algorithm=None, singular='singular_default')

Decompose zero-dimensional ideal self into triangular sets.

This requires that the given basis is reduced w.r.t. to the lexicographical monomial ordering. If the basis of self does not have this property, the required Groebner basis is computed implicitly.

INPUT:

• algorithm - string or None (default: None)

ALGORITHMS:

• singular:triangL - decomposition of self into triangular systems (Lazard).

• singular:triangLfak - decomp. of self into tri. systems plus factorization.

  • singular:triangM - decomposition of self into triangular systems (Moeller).

OUTPUT: a list \(T\) of lists \(t\) such that the variety of self is the union of the varieties of \(t\) in \(L\) and each \(t\) is in triangular form.

EXAMPLES:

sage: P.<e,d,c,b,a> = PolynomialRing(QQ,5,order='lex')
sage: I = sage.rings.ideal.Cyclic(P)
sage: GB = Ideal(I.groebner_basis('libsingular:stdfglm'))
sage: GB.triangular_decomposition('singular:triangLfak')
[Ideal (a - 1, b - 1, c - 1, d^2 + 3*d + 1, e + d + 3) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
Ideal (a - 1, b - 1, c^2 + 3*c + 1, d + c + 3, e - 1) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
Ideal (a - 1, b^2 + 3*b + 1, c + b + 3, d - 1, e - 1) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
Ideal (a - 1, b^4 + b^3 + b^2 + b + 1, -c + b^2, -d + b^3, e + b^3 + b^2 + b + 1) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
Ideal (a^2 + 3*a + 1, b - 1, c - 1, d - 1, e + a + 3) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
Ideal (a^2 + 3*a + 1, b + a + 3, c - 1, d - 1, e - 1) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
Ideal (a^4 - 4*a^3 + 6*a^2 + a + 1, -11*b^2 + 6*b*a^3 - 26*b*a^2 + 41*b*a - 4*b - 8*a^3 + 31*a^2 - 40*a - 24, 11*c + 3*a^3 - 13*a^2 + 26*a - 2, 11*d + 3*a^3 - 13*a^2 + 26*a - 2, -11*e - 11*b + 6*a^3 - 26*a^2 + 41*a - 4) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
Ideal (a^4 + a^3 + a^2 + a + 1, b - 1, c + a^3 + a^2 + a + 1, -d + a^3, -e + a^2) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
Ideal (a^4 + a^3 + a^2 + a + 1, b - a, c - a, d^2 + 3*d*a + a^2, e + d + 3*a) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
Ideal (a^4 + a^3 + a^2 + a + 1, b - a, c^2 + 3*c*a + a^2, d + c + 3*a, e - a) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
Ideal (a^4 + a^3 + a^2 + a + 1, b^2 + 3*b*a + a^2, c + b + 3*a, d - a, e - a) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
Ideal (a^4 + a^3 + a^2 + a + 1, b^3 + b^2*a + b^2 + b*a^2 + b*a + b + a^3 + a^2 + a + 1, c + b^2*a^3 + b^2*a^2 + b^2*a + b^2, -d + b^2*a^2 + b^2*a + b^2 + b*a^2 + b*a + a^2, -e + b^2*a^3 - b*a^2 - b*a - b - a^2 - a) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
Ideal (a^4 + a^3 + 6*a^2 - 4*a + 1, -11*b^2 + 6*b*a^3 + 10*b*a^2 + 39*b*a + 2*b + 16*a^3 + 23*a^2 + 104*a - 24, 11*c + 3*a^3 + 5*a^2 + 25*a + 1, 11*d + 3*a^3 + 5*a^2 + 25*a + 1, -11*e - 11*b + 6*a^3 + 10*a^2 + 39*a + 2) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field]

sage: R.<x1,x2> = PolynomialRing(QQ, 2, order='lex')
sage: f1 = 1/2*((x1^2 + 2*x1 - 4)*x2^2 + 2*(x1^2 + x1)*x2 + x1^2)
sage: f2 = 1/2*((x1^2 + 2*x1 + 1)*x2^2 + 2*(x1^2 + x1)*x2 - 4*x1^2)
sage: I = Ideal(f1,f2)
sage: I.triangular_decomposition()
[Ideal (x2, x1^2) of Multivariate Polynomial Ring in x1, x2 over Rational Field,
 Ideal (x2, x1^2) of Multivariate Polynomial Ring in x1, x2 over Rational Field,
 Ideal (x2, x1^2) of Multivariate Polynomial Ring in x1, x2 over Rational Field,
 Ideal (x2^4 + 4*x2^3 - 6*x2^2 - 20*x2 + 5, 8*x1 - x2^3 + x2^2 + 13*x2 - 5) of Multivariate Polynomial Ring in x1, x2 over Rational Field]

variety(ring=None)

Return the variety of this ideal.

Given a zero-dimensional ideal \(I\) (== self) of a polynomial ring \(P\) whose order is lexicographic, return the variety of \(I\) as a list of dictionaries with (variable, value) pairs. By default, the variety of the ideal over its coefficient field \(K\) is returned; ring can be specified to find the variety over a different ring.

These dictionaries have cardinality equal to the number of variables in \(P\) and represent assignments of values to these variables such that all polynomials in \(I\) vanish.

If ring is specified, then a triangular decomposition of self is found over the original coefficient field \(K\); then the triangular systems are solved using root-finding over ring. This is particularly useful when \(K\) is QQ (to allow fast symbolic computation of the triangular decomposition) and ring is RR, AA, CC, or QQbar (to compute the whole real or complex variety of the ideal).

Note that with ring=RR or CC, computation is done numerically and potentially inaccurately; in particular, the number of points in the real variety may be miscomputed. With ring=AA or QQbar, computation is done exactly (which may be much slower, of course).

INPUT:

• ring - return roots in the ring instead of the base ring of this ideal (default: None)

• proof - return a provably correct result (default: True)

EXAMPLES:

sage: K.<w> = GF(27) # this example is from the MAGMA handbook
sage: P.<x, y> = PolynomialRing(K, 2, order='lex')
sage: I = Ideal([ x^8 + y + 2, y^6 + x*y^5 + x^2 ])
sage: I = Ideal(I.groebner_basis()); I
Ideal (x - y^47 - y^45 + y^44 - y^43 + y^41 - y^39 - y^38
- y^37 - y^36 + y^35 - y^34 - y^33 + y^32 - y^31 + y^30 +
y^28 + y^27 + y^26 + y^25 - y^23 + y^22 + y^21 - y^19 -
y^18 - y^16 + y^15 + y^13 + y^12 - y^10 + y^9 + y^8 + y^7
- y^6 + y^4 + y^3 + y^2 + y - 1, y^48 + y^41 - y^40 + y^37
- y^36 - y^33 + y^32 - y^29 + y^28 - y^25 + y^24 + y^2 + y
+ 1) of Multivariate Polynomial Ring in x, y over Finite
Field in w of size 3^3

sage: V = I.variety();
sage: sorted(V, key=str)
[{y: w^2 + 2*w, x: 2*w + 2}, {y: w^2 + 2, x: 2*w}, {y: w^2 + w, x: 2*w + 1}]
sage: [f.subs(v) for f in I.gens() for v in V] # check that all polynomials vanish
[0, 0, 0, 0, 0, 0]
sage: [I.subs(v).is_zero() for v in V] # same test, but nicer syntax
[True, True, True]

However, we only account for solutions in the ground field and not in the algebraic closure:

sage: I.vector_space_dimension()
48

Here we compute the points of intersection of a hyperbola and a circle, in several fields:

sage: K.<x, y> = PolynomialRing(QQ, 2, order='lex')
sage: I = Ideal([ x*y - 1, (x-2)^2 + (y-1)^2 - 1])
sage: I = Ideal(I.groebner_basis()); I
Ideal (x + y^3 - 2*y^2 + 4*y - 4, y^4 - 2*y^3 + 4*y^2 - 4*y + 1)
of Multivariate Polynomial Ring in x, y over Rational Field

These two curves have one rational intersection:

sage: I.variety()
[{y: 1, x: 1}]

There are two real intersections:

sage: sorted(I.variety(ring=RR), key=str)
[{y: 0.361103080528647, x: 2.76929235423863},
 {y: 1.00000000000000, x: 1.00000000000000}]
sage: I.variety(ring=AA) # py2
[{x: 1, y: 1},
 {x: 2.769292354238632?, y: 0.3611030805286474?}]
sage: I.variety(ring=AA) # py3
[{y: 1, x: 1},
 {y: 0.3611030805286474?, x: 2.769292354238632?}]

and a total of four intersections:

sage: sorted(I.variety(ring=CC), key=str)
[{y: 0.31944845973567... + 1.6331702409152...*I,
  x: 0.11535382288068... - 0.58974280502220...*I},
 {y: 0.31944845973567... - 1.6331702409152...*I,
  x: 0.11535382288068... + 0.58974280502220...*I},
 {y: 0.36110308052864..., x: 2.7692923542386...},
 {y: 1.00000000000000, x: 1.00000000000000}]
sage: sorted(I.variety(ring=QQbar), key=str)
[{y: 0.3194484597356763? + 1.633170240915238?*I,
  x: 0.11535382288068429? - 0.5897428050222055?*I},
 {y: 0.3194484597356763? - 1.633170240915238?*I,
  x: 0.11535382288068429? + 0.5897428050222055?*I},
 {y: 0.3611030805286474?, x: 2.769292354238632?},
 {y: 1, x: 1}]

Computation over floating point numbers may compute only a partial solution, or even none at all. Notice that x values are missing from the following variety:

sage: R.<x,y> = CC[]
sage: I = ideal([x^2+y^2-1,x*y-1])
sage: sorted(I.variety(), key=str)
verbose 0 (...: multi_polynomial_ideal.py, variety) Warning: computations in the complex field are inexact; variety may be computed partially or incorrectly.
verbose 0 (...: multi_polynomial_ideal.py, variety) Warning: falling back to very slow toy implementation.
[{y: -0.86602540378443... + 0.500000000000000*I},
 {y: -0.86602540378443... - 0.500000000000000*I},
 {y: 0.86602540378443... + 0.500000000000000*I},
 {y: 0.86602540378443... - 0.500000000000000*I}]

This is due to precision error, which causes the computation of an intermediate Groebner basis to fail.

If the ground field’s characteristic is too large for Singular, we resort to a toy implementation:

sage: R.<x,y> = PolynomialRing(GF(2147483659),order='lex')
sage: I=ideal([x^3-2*y^2,3*x+y^4])
sage: I.variety()
verbose 0 (...: multi_polynomial_ideal.py, groebner_basis) Warning: falling back to very slow toy implementation.
verbose 0 (...: multi_polynomial_ideal.py, dimension) Warning: falling back to very slow toy implementation.
verbose 0 (...: multi_polynomial_ideal.py, variety) Warning: falling back to very slow toy implementation.
[{y: 0, x: 0}]

The dictionary expressing the variety will be indexed by generators of the polynomial ring after changing to the target field. But the mapping will also accept generators of the original ring, or even generator names as strings, when provided as keys:

sage: K.<x,y> = QQ[]
sage: I = ideal([x^2+2*y-5,x+y+3])
sage: v = I.variety(AA)[0]; v[x], v[y]
(4.464101615137755?, -7.464101615137755?)
sage: list(v)[0].parent()
Multivariate Polynomial Ring in x, y over Algebraic Real Field
sage: v[x]
4.464101615137755?
sage: v["y"]
-7.464101615137755?

ALGORITHM:

Uses triangular decomposition.

vector_space_dimension()

Return the vector space dimension of the ring modulo this ideal. If the ideal is not zero-dimensional, a TypeError is raised.

ALGORITHM:

Uses Singular.

EXAMPLES:

sage: R.<u,v> = PolynomialRing(QQ)
sage: g = u^4 + v^4 + u^3 + v^3
sage: I = ideal(g) + ideal(g.gradient())
sage: I.dimension()
0
sage: I.vector_space_dimension()
4

When the ideal is not zero-dimensional, we return infinity:

sage: R.<x,y> = PolynomialRing(QQ)
sage: I = R.ideal(x)
sage: I.dimension()
1
sage: I.vector_space_dimension()
+Infinity

Due to integer overflow, the result is correct only modulo 2^32, see trac ticket #8586:

sage: P.<x,y,z> = PolynomialRing(GF(32003),3)
sage: sage.rings.ideal.FieldIdeal(P).vector_space_dimension()  # known bug
32777216864027

class sage.rings.polynomial.multi_polynomial_ideal.NCPolynomialIdeal(ring, gens, coerce=True, side='left')

Bases: sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal_singular_repr, sage.rings.noncommutative_ideals.Ideal_nc

Creates a non-commutative polynomial ideal.

INPUT:

• ring - the g-algebra to which this ideal belongs

• gens - the generators of this ideal

• coerce (optional - default True) - generators are coerced into the ring before creating the ideal

• side - optional string, either “left” (default) or “twosided”; defines whether this ideal is left of two-sided.

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: H = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y})
sage: H.inject_variables()
Defining x, y, z
sage: I = H.ideal([y^2, x^2, z^2-H.one()],coerce=False) # indirect doctest
sage: I #random
Left Ideal (y^2, x^2, z^2 - 1) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
sage: sorted(I.gens(),key=str)
[x^2, y^2, z^2 - 1]
sage: H.ideal([y^2, x^2, z^2-H.one()], side="twosided") #random
Twosided Ideal (y^2, x^2, z^2 - 1) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
sage: sorted(H.ideal([y^2, x^2, z^2-H.one()], side="twosided").gens(),key=str)
[x^2, y^2, z^2 - 1]
sage: H.ideal([y^2, x^2, z^2-H.one()], side="right")
Traceback (most recent call last):
...
ValueError: Only left and two-sided ideals are allowed.

elimination_ideal(variables)

Return the elimination ideal of this ideal with respect to the variables given in “variables”.

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: H = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y})
sage: H.inject_variables()
Defining x, y, z
sage: I = H.ideal([y^2, x^2, z^2-H.one()],coerce=False)
sage: I.elimination_ideal([x, z])
Left Ideal (y^2) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {...}
sage: J = I.twostd()
sage: J
Twosided Ideal (z^2 - 1, y*z - y, x*z + x, y^2, 2*x*y - z - 1, x^2) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {...}
sage: J.elimination_ideal([x, z])
Twosided Ideal (y^2) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {...}

ALGORITHM: Uses Singular’s eliminate command

reduce(p)

Reduce an element modulo a Groebner basis for this ideal.

It returns 0 if and only if the element is in this ideal. In any case, this reduction is unique up to monomial orders.

NOTE:

There are left and two-sided ideals. Hence,

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: H.<x,y,z> = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y})
sage: I = H.ideal([y^2, x^2, z^2-H.one()],coerce=False, side='twosided')
sage: Q = H.quotient(I); Q #random
Quotient of Noncommutative Multivariate Polynomial Ring in x, y, z
 over Rational Field, nc-relations: {z*x: x*z + 2*x,
 z*y: y*z - 2*y, y*x: x*y - z} by the ideal (y^2, x^2, z^2 - 1)
sage: Q.2^2 == Q.one()   # indirect doctest
True

Here, we see that the relation that we just found in the quotient is actually a consequence of the given relations:

sage: H.2^2-H.one() in I.std().gens()
True

Here is the corresponding direct test:

sage: I.reduce(z^2)
1

res(length)

Compute the resolution up to a given length of the ideal.

NOTE:

Only left syzygies can be computed. So, even if the ideal is two-sided, then the resolution is only one-sided. In that case, a warning is printed.

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: H = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y})
sage: H.inject_variables()
Defining x, y, z
sage: I = H.ideal([y^2, x^2, z^2-H.one()],coerce=False)
sage: I.res(3)
<Resolution>

std()

Computes a GB of the ideal. It is two-sided if and only if the ideal is two-sided.

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: H = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y})
sage: H.inject_variables()
Defining x, y, z
sage: I = H.ideal([y^2, x^2, z^2-H.one()],coerce=False)
sage: I.std() #random
Left Ideal (z^2 - 1, y*z - y, x*z + x, y^2, 2*x*y - z - 1, x^2) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
sage: sorted(I.std().gens(),key=str)
[2*x*y - z - 1, x*z + x, x^2, y*z - y, y^2, z^2 - 1]

If the ideal is a left ideal, then std returns a left Groebner basis. But if it is a two-sided ideal, then the output of std and twostd() coincide:

sage: JL = H.ideal([x^3, y^3, z^3 - 4*z])
sage: JL #random
Left Ideal (x^3, y^3, z^3 - 4*z) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
sage: sorted(JL.gens(),key=str)
[x^3, y^3, z^3 - 4*z]
sage: JL.std() #random
Left Ideal (z^3 - 4*z, y*z^2 - 2*y*z, x*z^2 + 2*x*z, 2*x*y*z - z^2 - 2*z, y^3, x^3) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
sage: sorted(JL.std().gens(),key=str)
[2*x*y*z - z^2 - 2*z, x*z^2 + 2*x*z, x^3, y*z^2 - 2*y*z, y^3, z^3 - 4*z]
sage: JT = H.ideal([x^3, y^3, z^3 - 4*z], side='twosided')
sage: JT #random
Twosided Ideal (x^3, y^3, z^3 - 4*z) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
sage: sorted(JT.gens(),key=str)
[x^3, y^3, z^3 - 4*z]
sage: JT.std() #random
Twosided Ideal (z^3 - 4*z, y*z^2 - 2*y*z, x*z^2 + 2*x*z, y^2*z - 2*y^2, 2*x*y*z - z^2 - 2*z, x^2*z + 2*x^2, y^3, x*y^2 - y*z, x^2*y - x*z - 2*x, x^3) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
sage: sorted(JT.std().gens(),key=str)
[2*x*y*z - z^2 - 2*z, x*y^2 - y*z, x*z^2 + 2*x*z, x^2*y - x*z - 2*x, x^2*z + 2*x^2, x^3, y*z^2 - 2*y*z, y^2*z - 2*y^2, y^3, z^3 - 4*z]
sage: JT.std() == JL.twostd()
True

ALGORITHM: Uses Singular’s std command

syzygy_module()

Computes the first syzygy (i.e., the module of relations of the given generators) of the ideal.

NOTE:

Only left syzygies can be computed. So, even if the ideal is two-sided, then the syzygies are only one-sided. In that case, a warning is printed.

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: H = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y})
sage: H.inject_variables()
Defining x, y, z
sage: I = H.ideal([y^2, x^2, z^2-H.one()],coerce=False)
sage: G = vector(I.gens()); G
d...: UserWarning: You are constructing a free module
over a noncommutative ring. Sage does not have a concept
of left/right and both sided modules, so be careful.
It's also not guaranteed that all multiplications are
done from the right side.
d...: UserWarning: You are constructing a free module
over a noncommutative ring. Sage does not have a concept
of left/right and both sided modules, so be careful.
It's also not guaranteed that all multiplications are
done from the right side.
(y^2, x^2, z^2 - 1)
sage: M = I.syzygy_module(); M
[                                                                         -z^2 - 8*z - 15                                                                                        0                                                                                      y^2]
[                                                                                       0                                                                          -z^2 + 8*z - 15                                                                                      x^2]
[                                                              x^2*z^2 + 8*x^2*z + 15*x^2                                                              -y^2*z^2 + 8*y^2*z - 15*y^2                                                                   -4*x*y*z + 2*z^2 + 2*z]
[                 x^2*y*z^2 + 9*x^2*y*z - 6*x*z^3 + 20*x^2*y - 72*x*z^2 - 282*x*z - 360*x                                                              -y^3*z^2 + 7*y^3*z - 12*y^3                                                                                  6*y*z^2]
[                                                              x^3*z^2 + 7*x^3*z + 12*x^3                 -x*y^2*z^2 + 9*x*y^2*z - 4*y*z^3 - 20*x*y^2 + 52*y*z^2 - 224*y*z + 320*y                                                                                 -6*x*z^2]
[  x^2*y^2*z + 4*x^2*y^2 - 8*x*y*z^2 - 48*x*y*z + 12*z^3 - 64*x*y + 108*z^2 + 312*z + 288                                                                           -y^4*z + 4*y^4                                                                                        0]
[                                                  2*x^3*y*z + 8*x^3*y + 9*x^2*z + 27*x^2                                   -2*x*y^3*z + 8*x*y^3 - 12*y^2*z^2 + 99*y^2*z - 195*y^2                                                                -36*x*y*z + 24*z^2 + 18*z]
[                                                                           x^4*z + 4*x^4    -x^2*y^2*z + 4*x^2*y^2 - 4*x*y*z^2 + 32*x*y*z - 6*z^3 - 64*x*y + 66*z^2 - 240*z + 288                                                                                        0]
[x^3*y^2*z + 4*x^3*y^2 + 18*x^2*y*z - 36*x*z^3 + 66*x^2*y - 432*x*z^2 - 1656*x*z - 2052*x                                      -x*y^4*z + 4*x*y^4 - 8*y^3*z^2 + 62*y^3*z - 114*y^3                                                                        48*y*z^2 - 36*y*z]

sage: M*G
(0, 0, 0, 0, 0, 0, 0, 0, 0)

ALGORITHM: Uses Singular’s syz command

twostd()

Computes a two-sided GB of the ideal (even if it is a left ideal).

EXAMPLES:

sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
sage: H = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y})
sage: H.inject_variables()
Defining x, y, z
sage: I = H.ideal([y^2, x^2, z^2-H.one()],coerce=False)
sage: I.twostd() #random
Twosided Ideal (z^2 - 1, y*z - y, x*z + x, y^2, 2*x*y - z - 1, x^2) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field...
sage: sorted(I.twostd().gens(),key=str)
[2*x*y - z - 1, x*z + x, x^2, y*z - y, y^2, z^2 - 1]

ALGORITHM: Uses Singular’s twostd command

class sage.rings.polynomial.multi_polynomial_ideal.RequireField(f)

Bases: sage.misc.method_decorator.MethodDecorator

Decorator which throws an exception if a computation over a coefficient ring which is not a field is attempted.

Note

This decorator is used automatically internally so the user does not need to use it manually.

sage.rings.polynomial.multi_polynomial_ideal.is_MPolynomialIdeal(x)

Return True if the provided argument x is an ideal in the multivariate polynomial ring.

INPUT:

• x - an arbitrary object

EXAMPLES:

sage: from sage.rings.polynomial.multi_polynomial_ideal import is_MPolynomialIdeal
sage: P.<x,y,z> = PolynomialRing(QQ)
sage: I = [x + 2*y + 2*z - 1, x^2 + 2*y^2 + 2*z^2 - x, 2*x*y + 2*y*z - y]

Sage distinguishes between a list of generators for an ideal and the ideal itself. This distinction is inconsistent with Singular but matches Magma’s behavior.

sage: is_MPolynomialIdeal(I)
False

sage: I = Ideal(I)
sage: is_MPolynomialIdeal(I)
True

sage.rings.polynomial.multi_polynomial_ideal.require_field

alias of sage.rings.polynomial.multi_polynomial_ideal.RequireField